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My answer is regarding the first part of your Question 1. There is a paper linked in comments (below your question) which shows that, in general, there is no winning strategy. The paper is based upon essentially assuming that any given sequence of tetrominoes can occur.

The rules for what sequences of tetrominoes are or aren't allowed in a game is usually called a "Randomiser" (in terminology of the game). A randomiser that allows any sequence of tetrominoes is called "memoryless". So the paper shows that given a memoryless-randomiser, there is no general winning strategy. However, the analysis might not apply to games(even otherwise assuming exactly same game mechanics as in paper) which implement a different randomiser because certain sequences of tetrominoes are forbidden with probability 1 (at any possible point in the game).

It has already been hinted in the other answer that the winning strategy for single player with, what is called a bag randomiser, and some further rules/features has been known for almost a decade (see for example one of the answers: https://math.stackexchange.com/questions/1135388/an-impossible-sequence-of-tetris-pieces). It should be mentioned that some aspects(I think) have been improved upon for this loop in recent years.

But, at any rate, what is surprising for me is that there doesn't seem to be much work done beyond that. Considering "just" the single-player gravity based variant of endless tetris there are three main aspects (keeping it as short as possible):

(1) Basic Features: These include, among other things, features such as "Hold" and "Number of Next Piece Previews".

(2) Gravity Rules: There are two main choices "0G" and "20G". 0G or very close to 0G occurs in most commercial games. 20G (instant drop) occurs in the well-known TGM series. With 0G one can decide whether to include "softdrop" or not. What that means is that when a tetromino falls on the surface it can be moved around on the surface (of the stack), before finally locking in (the proof I linked actually seems to use that in a couple of situations). With 20G there are certain basic "kick-rules" (the full explanation of rules would be little longer but they are well-documented).

(3) Randomiser: Randomiser, as has been mentioned, desrcibes rules for which sequences of tetrominoes are or aren't allowed. In actuality, this is done to reduce probability of unfair situations. As far as I know, "7 piece bag" is the most common randomiser for commercial iterations (atleast for some time now). However, there is no shortage of other randomisers (some with very basic and short descriptions) that have been used in fairly noteable fan-made or commercial games.

Note that I have abridged the post enormously (leaving out discussion of myriad of modes that might be interesting for analysis) just to emphasise a somewhat more focused question about which it seems not much is known: " Suppose we set certain rules of play regarding the falling tetrominoes -- coming under points (1) and (2) in my description above. Given a certain randomiser, how do we find out that endless play is possible in general or not? " It would be interesting to see some progress on this question regarding reasonably large classes of randomisers.

P.S. Randomisers also include probabilistic considerations with them, which I glossed over (which might only be relevant if we are interested in questions beyond just a guaranteed winning strategy). For example, the more strict definition of memoryless-randomiser would be that at "any" point in the game, regardless of what finite sequence of tetrominoes was handed over, the probability for every tetromino(to be the next one) would be 1/7 (uniform across tetrominoes).