What modal verbs do natives use nowadays?

The question is to some extent a matter of opinion. However, I would say that your examples are not examples which are widely considered outdated, especially in written English.

We must is used in formal speech and in writing. In informal speech, it has been largely replaced by we need to, we have to and we've got to.

Gotta is slang for have got to, and the latter is certainly not outdated. We don't gotta go yet is terrible English, and I suggest you do not get into the habit of saying something that will make you stand out as a poor speaker of English.

The list of modal verbs in use today is too broad to be addressed here, but you can learn them by reading. Reading personal blogs will give you an idea of how people speak. Reading articles will give you an idea of how people write today.

We must -> We have to

These are both in common usage still - as a native speaker I would use "must" for emphasis, or to talk about things we should consider doing rather than absolutely are required to do. For example

"We must get around to getting you that new dress..."

In this context "have to" wouldn't work.

Incidentally note the difference between two pronunciations of

"a list of things we have to do in our holidays"

pronounced [hafta] it is obligation, pronounced [havtu] it is a matter of possibility (What things do we have to do at Easter camp? We can go fishing, sailing, rock climbing, ...)

Another related one

"she was supposed to have killed him"

where the voicing on supposed makes the "thought" meaning and the devoicing the modal meaning. The dropping of segments including voicing and differences in place of articulation (gotta, haft, sposta) is part of the entropic pressure on language, and it is those things that are becoming common frequent idioms that are most affected. It is the same process that lead to "the day" -> "to day" -> "to-day" -> "today" and "the morrow"... (note cognate Morgen in German still means both morning and tomorrow: in the morning = tomorrow). Also I see the same affect in "one of" -> "one off" and "would have" -> "would've" -> "would of" -> "wouldf".

May I come in? -> Do you mind if I come in?

Actually I am not that polite or pedantic, and would say "Can I come in?" [knai kmin]. All are quite reasonable, your two are both polite, but my version is the most natural.

We don't have to leave yet. -> We don't gotta go yet.

I wouldn't say either, but would certainly use "have to" or "need to" rather than "don't gotta". I would also say "I've got to go now" or "I have to go now" with almost equal likelihood, the former more informal but I could still say either in an important meeting or phone call (and have), as well as "I really must go now".

In terms of what is going on here, in every community, in every new generation, language is changing and new idioms are emerging (like the many uses of "like"). But the international media, and in particular movies and TV shows, means this spreads and we no longer have the insular effects to the same degree.

In practice, it is important to learn the new modals "hafta", "gotta", "sposta", "kinda", "like" if you want to fit in and sound natural/native. The last two though I wouldn't use in a formal context.

If someone says using 'gotta' as proper English, as a rule of thumb, I would recommend doing the exact opposite of everything they say. Gotta is phenomenally terrible English.

Seriously, every recommendation that you've listed is wrong.

• We must ➙ We have to

These are similar but they say different things. Many people will treat must as a stronger form of have to. They aren't simply interchangeable.

• May I come in? ➙ Do you mind if I come in?

This is just a completely different way of wording the same sentiment. If you were trying to pad the word count of your essay, or if you were trying to be extremely formal, then use the second. Otherwise, the first one is perfectly fine. Actually, it is better English than most people would use. Most people would say "Can I come in?". Using "May I" is technically more correct, but it isn't often used anymore.

I was always taught that 'Can I' is a question of capability. "Can I jump ten feet in the air?". Whereas 'May I' is a question of permission. Whenever someone would ask "Can I go to the bathroom?" one of my high school English teachers would always respond "Well of course you can go but, no, you may not go."

• We don't have to leave yet. ➙ We don't gotta go yet.

No. Just No. I don't know if words exist to describe exactly how bad that is.

In addition assuming the opposite of whatever this person recommends, I would advise against learning english from an American. According to Wikipedia only 60% of Americans are literate. And of those who are literate, most can only read at the 7th grade level.