Is it that in the UK, there could be "pain and suffering"
independently of another tort, while in the U.S. it could take place
only in connection with some other offense (e.g. the broken vase)?
The whole question is too much for me to take on (there are significant differences in the law of non-economic damages between the U.K. and the U.S. beyond those discussed below), but I'll answer this part.
The short answer is "yes," there could be "pain and suffering" awards (more often and more accurately called "non-economic damages" awards) in the U.S. that are not in connection with another offense that cause physical harm to the person bringing the claim or their property, even though defamation liability is more narrow in the U.S. than in the U.K. Non-economic damages may include pain, emotional anguish, humiliation, reputational damage, loss of enjoyment of activities, or worsening of prior injuries; in some states, they are referred to as pain and suffering.
The U.S. recognizes a common law tort known as "intentional infliction of emotional distress" a.k.a. "outrageous conduct", that can award damages for purely non-economic harm that is independent of any physical harm to anyone. This has to be a course of conduct intentionally calculated to utterly crush someone emotionally without justification, generally through some form of humiliation. The nature of the conduct must far exceed what would ordinarily be considered harassment or taunting. There is also a closely related concept called a prima facie tort which involves intentional and unjustified harm to others not covered by another recognized tort, which is much less widely recognized.
The U.S. also recognizes a common law tort known as "negligent infliction of emotional distress" which applies to allow a non-economic damages recovery from someone in the "zone of danger" of an accident that causes physical injury to another person or to property. The classic case would be brought by a parent holding hands with a child whose child is hit by a car and grievously injured or killed right before their eyes. This tort is not very common and not accepted in all jurisdictions.
Closely related to negligent infliction of emotional distress damages are awards for loss of consortium, which is a claim for non-economic harm caused to a spouse by tortious conduct that injures the spouse by the non-injured spouse directly against the wrong doer by the non-injured spouse (in some states a parent can also make a loss of consortium claim with regard to a child).
Also somewhat related is the concept of a wrongful death lawsuit, which is typically vested by statute in the next of kin of the person wrongfully killed, rather than in the decedent's probate estate, and can include both economic and non-economic damages. Often in a wrongful death case, rather than seeking economic damages, one can seek a flat compensatory amount fixed by statute called a "solatium" that is primarily representative of non-economic harm to the next of kin.
There are also other U.S. torts which can give rise to non-economic damages without physical or economic harm to person or physical property.
Non-economic damages without physical harm are frequently awarded in civil rights lawsuits brought against someone acting under the color of law pursuant to 42 U.S.C. § 1983 even when the right violated, such as a right to vote, or a right to use public accommodations in a manner not subject to discrimination, does not involve damage to person or property. Similarly, a violation of a discrimination or sexual harassment statutes, such as the federal Civil Rights Act of 1964, by a private individual can give rise to a claim for non-economic damages even without proof of economic damages.
The U.S. also recognizes some privacy torts (not all of which are recognized in every state) and claims for breaches of fiduciary duty, for which there may be a cause of action for non-economic damages even in the absence of proven economic damage. For example, if an attorney breaches his duty of confidentiality to publicly reveal highly embarrassing personal facts about you, you could sue the attorney for non-economic damages.
Claims for nuisance (e.g. from a neighbor's noisy or smelly conduct on the neighbor's own property) can give rise to claims for non-economic damage, and while they would often include claims for related economic damage (often quantified in the form of lost rental value of property), this isn't always required.
Most U.S. states have abolished their heart balm torts such as alienation of affection (i.e. causing someone to divorce their spouse), "criminal conversion" (i.e. adultery), and also torts related to broken engagements to marry, but a few states, most notably, North Carolina, retain them and essentially allow private adultery lawsuits for both economic and non-economic harm (or one without the other).
Claims for libel per se, which involve a false statement about another which accuses him/her of a crime, immoral acts, inability to perform his/her profession, having a loathsome disease (like syphilis) or dishonesty in business, can give rise to a claim for non-economic damages, in the absence of any proof of specific economic harm arising from the related damage to your reputation.
The U.S. does not recognize defamation lawsuits based upon truthful statements, even though, under many historical criminal defamation statutes, statements tending to harm the reputations of the dead, and statements calling attentions to the natural deficiencies of the living (i.e. mocking mentally or physically disabled people) were actionable. Under U.S. law, freedom of speech protects those statements.