Learn English – Use of the phrase “having to”

It is probably a matter of definition, not of true inherent "belonging to". I believe the etymology of this kind of "to" is the usage of a preposition before the infinitive, in order to indicate a relation of direction or purpose between a finite verb and the infinitive: I went to school; I went to pick him up. This probably originated in predicates with verbs that have a direction. But that was long ago, when the infinitive still had a distinctive form in English or Proto-Germanic; I think it was something like *eatan (don't pin me down on this, I haven't looked it up). Later, when the infinitive became indistinguishable from other forms of the verb, this to evolved into a more general marker of the infinitive. So the most precise definition would probably be to say that to belongs neither to the finite verb ("like") nor to the infinitive.

That said, you could say "like often goes with to, so we say to belongs to like", or "to often introduces or links to infinitives, so we say it belongs to the infinitive". In lists of phrases, educational books often tend to emphasise the connection with like; in passages that describe the infinitive, they tend to treat it as part of the "full" infinitive, as opposed to the "bare" infinitive in "I will go". I believe this is the most traditional way to describe it.

You don't normally use Present Perfect with stative verbs (have, be, like, seem, prefer, understand, doubt, know, etc.) Here's a longer list - in general, they apply to states that last for some time.

In some contexts, such as "How long have you had/been having these symptoms?", there's no real difference. Arguably, been having calls more attention to the fact that you're still having the symptoms, but I doubt many people would consciously either make or hear that distinction.

A "rule of thumb" for to have is: when it means to experience, you might want to use Present Perfect; when it means to own, you almost certainly don't.

Here's an example for to be using the "slightly unusual" Present Perfect in a construction which is perfectly valid, and is probably the most succinct way of expressing the intended meaning...

By now the new cook will have been being introduced to her duties for several weeks.