Learn English – How to use “happen to” like this

You are mistaken in supposing this to be a recent innovation: to the best of my recollection it arose in the late 1950s. At that time it was supposed to be characteristic of the ‘beat’ community, but it entered youthspeak almost immediately and has been current among the young ever since, at least in the United States (I cannot speak to other dialects). As teenagers grow older and are called upon to address more diverse audiences more formally, they generally learn alternative expressions (as it were, so to speak, I think, in my opinion) culled from severer registers; but like never entirely disappears from speech with close friends.

It has a useful and indeed essential function in speech as a discourse marker. When it first arose it was employed primarily as an anticipatory emphatic, to signal that what follows will be of particular importance—Maynard G. Krebs’ favorite expression was “Like, wow!”—but by the end of the 60s it had acquired additional anticipatory uses: notably 1) as a signal that a subsequent pause will not mark the end of an utterance but further processing and 2) as a lexical ‘quotation mark’.

There is thus nothing to deprecate in the usage except snobbish preference for more prestigious markers. If the usage annoys you, I suggest you seek a more pretentious class of interlocutors.

I have to disagree. Grammatically, the sentence is doing no more and no less than asking to what extent the other person likes the shirt.

Compare: How do you like your boss?

This does not imply any favor or disfavor on the part of the person asking.

That said, practically speaking, when you ask someone about a shirt, you are typically considering buying/wearing it, and thus typically have at least a semi-positive opinion of it. But that's true of all your example sentences.