The modal verbs are a subset of the "verbs of being", which are properly called auxiliary verbs. The classical modal verbs are:
- shall
- should
- will
- would
- may
- might
- must
- can
- could
Modal verbs are peculiar in that they have no infinitive form (you can't say to shall) and cannot be combined with other modal verbs in Standard English (you can't say I will might go).
By way of contrast, the other auxiliary verbs are formed from be, have, and do, and they don't have these properties. You can say to be, and you can combine a non-modal auxiliary with a modal: I should have bought those shoes.
For more detailed discussion, see Wikipedia.
The extent to which we can still call these preterite-present, is that they have present-tense uses for the past tense form. For example, while we can use could as a straight-forward enough (if irregularly formed) past tense of can in a sentence like:
I could run that far when I was younger, but I can't any more.
We can also use could in the present tense.
I could run that far when I was younger, but I *could*n't any more.
Including in forms where we must use could, and can wouldn't serve:
I could really do with a coffee right now.
Remember, that can and could are forms of the same word. You said "I don't see how can, will, may and shall express a past action or state", and the answer is rather than the past tense of those words—could, would, might & should can apply to the present tense.
Now for the history bit!
In Old English there were two main forms of verb; strong and weak. They differed in terms of how they were inflected for person, tense, number and mood. Indeed, we still have the legacy of this in Modern English, so some people still use the terms for Modern English.
A weak verb would change tense by having a set ending added to it. This is the case with most Modern English verbs (walk → walked, watch → watched, auction → auctioned).
A strong verb would have a change in vowel to change tense and sometimes mood. This is still the case of some of the Modern English equivalents (ring → rang/rung, bite → bit/bitten, come → came, shake → shook/shaken).
Because there are relatively few strong verbs in Modern English, many people don't use the terms weak and strong about Modern English, but just regular and irregular, considering all the remaining strong verbs as special cases.
The two also differed in how number and person worked with each type. There was a third class that would act like the strong for the past tense and the weak in the present tense, in this regard.
The fourth class were the preterite-present verbs, where a form you would normally have with a strong past tense, was used in the present tense, and then the other inflections were done as if they were weak. So for this reason, they had some qualities of past-tense verbs when they were in the present. Hence the contradictory name preterite-present.
Old English had many more preterite-present verbs than Modern English. The only remnants in Modern English are the auxiliaries. This was not the case in Old English in both directions; by which I mean that not only where many other verbs preterite-present, but that the auxiliaries weren't auxiliaries in Old English (they also weren't defective, you could say sculan in Old English while the closest translation "to shall" or "to should" isn't correct Modern English.
Willan was neither defective, nor preterite-present, though the modern will/would is both. It was a strong verb.
Best Answer
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.