I used to be a litigator in civil rights cases against the police for misconduct against civilians: that personal experience is the basis for my discussion herein. I practiced for 11 years before quitting the profession.
The Fourth Amendment search and seizure protection is basically dead except for white collar crime and entry into the home. Your situation involves one of the areas where the 4th Amendment still imposes some limitations on police action. Such protection is always incomplete and contingent. If the police really want to enter a home they can and will do so, and they can and sometimes will use deadly force to overcome resistence. The fourth amendment does not barricade doors or make someone bulletproof.
Police generally are not held accountable for violations of civilian rights, in the absence of substantial and sustained political pressure. You cannot sue and win money damages for the mere violation of a civil right; you must show that the violation caused economic, physical or mental harm that can be proven, and are quantifiable and compensable without getting too theoretical. In federal courts police have near-total immunity to suit because of a judge-made loophole in the civil rights law called qualified immunity. That rule states that police are immune to suit unless they violated a clearly established consitutional right and "no reasonable police officer" in the officer's circumstanmces would have thought the officer's conduct was lawful. JUdges decide these issues, not juries, and they can do so at any stage in the case if they base their decision on facts that are not in dispute. Courts love doing this. The constitutional right is not defined at a general level ("right to be free from warrantless entry"), but with highly specific reference to the facts of the case. Almost all constitutional law is left unsettled and ambiguous by the judiciary even at the general level; it is very difficult to make the case that a particular fact pattern gives rise to "clearly established" rights. What a reasonable police officer is, or whether any such officers exist, has never been defined. This hypothetical officer is basically a vessel to be filled with judicial bias (both individual and institutional). Courts usually have little interest in evidence of best practices or police department policy: the cases say that an officer can violate policy, procedure, etc. and still pass the reasonable officer test.
Qualified immunity kills most claims against individual officers. A case called Monnell kills most cases against the municipality or police department. That case says that you cannot sue the agency or city just because they employ an officer that committed a tort against you. You Have to show that the violation of your rights was the result of police following an unconstitutional policy or widespread and officially-tolerated practice. Police departments therefore do not create official policies or shield rulebooks from disclosure as classified. And the courts will throw out nearly anything you bring as evidence of a policy and practice short of videotaped confession by the mayor. On the criminal side, they police have surprisingly little interest in what happens in criminal cases that arise out of their arrests and investigations. Police officers are rewarded or punished internally based on whether they meet (illegal) quotas on writing tickets and making arrests. Whether charges are filed or not by the DA, and anything after that, has no effect on them in their jobs. If a judge finds out that a cop committed misconduct or perjury, the judge will do nothing. (This immunity from consequences prevails, unless there is substantial and sustained political pressure, of a kind that is rare enough that police can safely ignore the possibility in their day-to-day work). SO, when you ask the question: "Can the police do x y or z," the answer is generally yes, whatever you are referring to. They can do whatever they want because there really aren't any powers that oppose them or any systems in place to limit their conduct. Civil unrest is really the only thing that limits police conduct.
Take note that neither the probable cause test nor the qualified iummunity test give any consideration to what the police actually believe or what they want to accomplish. Whether there is probable cause is determined by what a reasonable police officer would conclude based on the facts objectively available. Police may have bad or racist or even illegal motives in taking action -- the courts do not want to know about it and will not entertain arguments of that kind, as a general rule. Generally, facts act as a one-way ratchet: a fact that is not obviously absurd or baseless is usually considered something on which a "reasonable police officer" can rely. The existence of other facts negating probable cause usually makes no difference unless they are either totally overwhelming or totally indisputable (video of an event is not indisputable). There is no duty to investigate to determine the true facts before taking action. There is no legally-enforceable duty to perform any police functions, including protecting citizens from crime. Thus, doing nothing is a VERY low risk move for police except under very unusual circumstances. The police will freely invent facts to support their official conduct: perjury is a fundamental tool of modern policing, as indispensible as the radio and the gun. To repeat, judges and district attorneys who learn that a cop has committed perjury will do nothing. I would have no fear betting my life that a police officer could commit perjury 100 times without being punished once.
So, unless you are specifically addressing the question of the admissibility of evidence obtained in an entry/search in a criminal matter your question really should not be: "Is there probable cause?" it should be: "Do the police want there to be probable cause?" If the answer to the second question is yes, then the police will almost certainly enter.
Generally, the police are interested in maximizing their overtime hours while doing as little real work as possible. Something that doesn't lead to extra overtime pay, doesn't get them points towards their quota, and which involves effort of any kind, is going to be a very low priority for the average cop. This includes stopping a crime in progress. However, some police have emotional investment in being in control of their territory. Something they see as a threat to that control will meet their vigorous and often physical opposition, even if the thing threatening their control is perfectly legal. In addition, police can be stirred to action by reputational concerns. WHile having a bad reputation (individually or collectively) poses no real threat to them, most police have some concern for their reputation to the extent that it may affect their social status. In high-visibility situations, police may either change their conduct to be more in line with public expectations, or take steps to conceeal their conduct or confuse onlookers.
So, just because a violent crime is alleged to be in progress, does not mean the police will want to do anything about it. Will it bring overtime? WIll handling it be easy? If its a violenmt crime, yes to the first question and no to the second. It could go either way. If the victim is a disfavored minority, this is less liklely to trigger a territorial defense response than if the victim is someone the officers see as within their sphere of protection. Police repsonse to domestic violence crimes is extremely variable, but disinterest often wins out because they are difficult and dangerous to handle, and the police see them as a problem which police action usually can't solve (and this is not without reason).
Most police forces have task forces of various kinds (swat, drug task forces) which get special training, and more importantly, higher status and cool militaristic equipment. Many of the officers on these task forces enjoy the excitement of kicking down doors, etc., and they will take action in a wider variety of situations, even if the factors that motivate most police officers are absent. So the same scenario can go very differently depending on whether it is regular uniforms or task force guys who show up first.
So, whether the police will want probable cause to exist will depend on whether the incident presents motivations or disincentives of the kinds mentioned above, and it will depend in part on who the police are who are available to respond.
Just so you have it clear, the 4th Amendment rule on entry to the home is this: If the police have a warrant they can enter according to its terms, without no need for any additional facts or circumstances to be present. If they don't then they need BOTH probable cause and exigent circumstances. Exigent circumstances means some kind of an emergency. It isn't only the sort of emergency where people might come to harm: the imminent destruction of evidence can also be an exigent circumstance. Again, police who want to enter will help themselves by inventing additional facts to meet the standards. Police who don't want to take action are perfectly capable of ignoring a truckload of exigent circumstances. Police are also good at using lies (which is legal) and threats (sometimes legal, sometimes not) to induce the person to allow them to enter, at which point civil rights are no longer an issue. Because the cop's level of motivation is a major factor in determining what happens, there is often inconsistency in how similar incidents are handled at different times by different officers.
The thing that the forth amendment protects is not your right not to have cops barge into your home, it is your right not to have the government build a case against you through fishing expeditions inside your home. Whatever the Constitution may say, the only real recourse the householder has is, in their criminal trial, to demand and possibly obtain exclusion of the evidence obtained in violation of their rights. It can take years to win a concession that your rights were violated. Many people spend those years in prison. The court may also agree that your rights were violated but allow the evidence anyway, for a variety of reasons, most of them disgraceful.
JUst to wrap things up, state and local officers are subject to the limits of the consitituion, at least theoretically, and criminal proceedings must, in most aspects, meet federal constitutional standards. These standards are low. State/local officers and state courts are also bound by their state constitutions. All states have rights analogous to the bill of rights written into their constitutions. Most leave it at that, and most state judiciaries take an active interest in making sure their constitutional law is basically the same as the federal. However, there can be differences, and if there are differences, they have to be in favor of the defendant. On the civil side, most states have rules similar to qualified immunity and Monell, making it very hard to sue police. Some, in addition, have limitations on damages which make even "successful" cases hardly worth the trouble. That said, a warrantless entry is a fact pattern where some legal protections still exist, where there is a non-hypothetical (small but greater than zero) possibility of success in a civil suit or a motion to exclude criminal evidence. It is one of the few fact patterns that can get even conservative judges angry at the police/municipality (occasionally).
In your proposed plot, I would expect that the police would have trouble determining in real time which calls were illegitimate. The doors they kick down usually belong to people with no social status, and police calibrate their conduct accordingly. Thus, the police could find themselves in unfamiliar and uncomfortable circumstances if they were uncertain whom or what they might find at a 911 address. Most would probably err on the side of caution, and this could result in them neglecting a real emergency. With asympathetic non-minority victim, such a scenario could generate enough political pressure to get their attention, if not to threaten their security. I think they would pretty quickly tire of the extra work, reputational risk, and interference with their territorial control, and would make stopping the crank caller a high priority. Police departments do not know how to investigate crime, which would be a problem for them, but manpower can substitute up to a point for institutional failure to prioritize investigative capabilities. Some task forces have officers of skill and training (often military training), and it would probably also be possible to get the feds involved in a case like that, if they were so inclined.
I hope this helps.
