Yes, it's generally fine to mix forms like that. Parallelism might be good sometimes, but native speakers aren't always consistent about that.

Some other complications here are:
1. Repetition of auxiliary and lexical 'have'. Because repetition is often avoided, "I've had to" might be less likely than, for example, "I've needed to".
2. Lexical aspect (Aktionsart). The perfective (and other aspects like progressive) are often more relevant for non-stative verbs. The difference between "I've had to" and "I had to" is minimal (and "*I was having to" is ungrammatical), but it's more different with lexical verbs like "eat", as in "I've eaten", "I ate", "I was eating".
3. There's also the issue of habituality: the simple past tends to (like the present) represent ongoing states or habits, rather than specific action events. The perfect is more likely with an action.

In the end, it's all flexible, and the differences are fairly small. But if you change the sentences or verbs you might find some of those differences become a little more important.

1. Can you give me an example of what you said here? I'm not sure if I understand.

Speakers are more likely to avoid saying something that has the same word repeated twice in a row. There's no rule here. It's just a tendency.

2. Can you tell me what this minimal difference between "I've had" and "had to" is in this particular context I provided? I'll be more aware of which form to use in the future.

No, it really doesn't matter. But part of that is because of the particular verb you chose. It might matter a little more (and even then not very much) with another verb.
Roughly, I suppose you could say that the past tense describes the past, while the perfect expresses something about the results of the past. But that's basically the same thing. They're not really contrastive here. If you asked with sentences containing another verb, especially one that doesn't so easily permit a habitual reading, I might be able to say something else.

3. So, I can introduce some new information in the present perfect (as in example 3), and then switch to the past tenses (e.g. simple or continuous), and it will be correct grammatically and semantically?

Oh, I think I forgot to reply about that sentence above. My numbers 1, 2, 3 were NOT referring to your sentences, just general comments (three of them, about your first two sentences).

Regarding your third sentence:

Also, I'd like to make sure if it's correct to combine the present perfect and the past simple in one sentence, like:
3. I've watched the film you recommended, and I really liked it.

Yes, that is correct. It would be very strange sounding to say "I have really liked it". It's hard to explain why, but it has to do with the specific verb choice.

It's possible to say something like:
"I've enjoyed/liked going to movies, but now I don't like to do that anymore."
The emphasis here is on (1) how it is complete (perfective) with a change now; and (2) how it focuses on the resulting experience rather than the event itself (backgrounding).

But that kind of usage is very rare.

In short, perfect forms are often used to indicate something leading up to something else, and in that way they're often or usually mixed with simple tenses.

Yes, that sounds about right.

In (1) you're saying that you now know things, and in (2) you're saying that you learned them (probably yesterday). Same effect, different focus. Sort of like passive vs. active sentences, but even more subtle.

By the way, from your questions here, if you want to know more about verbs, I really would recommend reading about "lexical aspect" (or Aktionsart):
http://www.glottopedia.org/index.php/Aktionsart
https://en.wikipedia.org/wiki/Lexical_aspect

There's a lot to read about that, but when dealing with issues of (grammatical) aspect in usage, especially interaction between two verbs, and it's a complex topic that a lot of people have written about, but it's also relatively easy to follow if you understand the basics, and there's just a lot of information about specific details.

Either way. There isn't anything "correct" or "incorrect" about that. It just frames it a little differently.

Yes, that sounds accurate. (I would say that the past tense would be best for story telling, recounting events to someone, disconnected from the present, rather than, as you say, knowing something now.)

Yes. (But again "correct" is the wrong word.)