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.