So, lets say for argument's sake the search is illegal (we'll discuss scenario later).
Is the murder weapon inadmissible?
No. Evidence seized in an illegal search is inadmissible for use in court, as to allow it would reward the state for breaking the law. At time of arrest it's still evidence and depending on what other evidence comes out might not be inadmissible (More on this later). If the knife is the only thing linking the driver to the murder, then it's inadmissible.
What would happen to the driver, who was most likely the murderer?
At this stage, the driver might be arrested and charged... inadmissibility is a matter for pre-trial motions, which takes place after the cop made the search. Whether the guy is a murder or not, he is a suspect who has been accused.
Can the police arrest him, and investigate him with the hope of finding other evidence that would be enough to sentence him for the murder, despite the inadmissibility of the actual weapon?
This all takes place before the inadmissibility of evidence is declared, so they certainly could investigate the guy, cuff him, read him his rights, book him, Danno, and put him in jail to help build their case. HOWEVER, if the knife is the only thing linking the man to the crime, then all of this becomes what's called "Fruit of the Poisonous Tree". Because the knife was used as the sole evidence to secure search warrants against the suspect, and should the knife become inadmissible, anything that resulted from the execution of the warrants is itself inadmissible.
is the murderer in practice immune from being charged for this case, since the weapon cannot be tied to him?
Not entirely. Rarely is a murder weapon needed to secure a conviction. Not only that, but while the search is illegal, there is an exception called "Inevitable Discovery" which holds that evidence seized illegally initially may still be admissible if the cops can show that the legally obtained evidence would have led to the knife. Suppose they had CCTV footage and captured the man entering the area of the scene and found out he had a connection to the victim, and got a warrant that would have included his car... that would mean the knife is admissible even if the search was illegal.
If the weapon is considered inadmissible, is it returned to the suspect? If yes, would it also be returned even if it is of a type that the suspect is not legally allowed to own?
Yes to both, though when it would be returned is subject to possible use in other investigations. Additionally, while I don't know of any knife ban laws in the U.S., there are examples of contraband seized by arrest for a different offense was not returned when the initial offense was overturned on appeal, meaning the contraband evidence was no longer validly seized and thus that half of the case was overturned... but the defendant wasn't returned his drugs.
Is the weapon permanently inadmissible for all future, or could it be used in a later court proceeding?
No. I'm starting to suspect I know the TV show and episode that inspired this question, but the search violated suspect 1's rights and thus was inadmissible for his trial. However, if a second suspect was discovered as the evidence was developed, and the investigation lead to the knife (say... suspect 2 had access to suspect 1's car... like say... through his job at a car wash that suspect 1 was patronizing... and planted the knife in suspect 1's car to frame him) then the knife is admissible because of inevitable discovery rule.
Now, the TV Show I alluded too, and you'll forgive me as it's been a while, the events were that the cop pulled over suspect 1 for a valid reason (busted tail light, intentionally done by suspect 2 to get attention) and saw the knife on the backseat of the car, which is not a violation of search and seizure rules. The bloody knife was in plain view and gave probable cause to arrest and search the entirety of the car. If you leave evidence in a place where the officer can see it, they can seize it in a car at least. Places like the glove box, under the seats, or the trunk would properly hide it and not allow the officer to search the car, but through the windows is just fine.