Yes, that is in accordance with section 5.1 of the Tetris specification. https://www.colinfahey.com/tetris/tetris.html

The definition of "Standard Tetris" is an idealized model of the most important characteristics and behaviors of the first IBM-PC implementation of the Tetris game (circa 1986-1988). The idealized model is based upon inferring the apparent intentions of the developers of the first IBM-PC implementation of the Tetris game. For example, it seems reasonable to infer that the developers of the first IBM-PC implementation of the Tetris game intended to select the shape of each new falling piece "randomly", and that the use of the Borland C implementation of the rand() function was merely a practical approximation of the intention. The definition of "Standard Tetris" specifies that the shape of each new falling piece is to be selected "randomly". This ideal behavior cannot be achieved by any implementation, but implementations can approximate the ideal behavior.

Yes, so much research in Computer Science has been based on Tetris that someone found it worthwhile to create a specification for the game.

You're probably thinking of the "Random Generator" which is used by the Tetris Company for new Tetris games around 2001 or so. Previous Tetris games did not necessarily use this system (I know the NES and Gameboy versions certainly did not) and the other popular variant, Tetris Grand Master, has its own system. Sources below:

https://tetris.wiki/Tetris_Guideline#See_also

https://tetris.wiki/Random_Generator

https://tetris.wiki/TGM_randomizer

Not random; there's no notion of randomness here. It's worst case analysis in a Tetris board that is provided in the input to the algorithm (i.e. with input length proportional to its size, as opposed to specifying integers corresponding to its dimensions, which will give a not-easier computational result), alongside a sequence of tetrominos; the question is to devise a strategy of player moves achieves a standard objective; e.g. the maximum number of rows cleared, or achieves the objective to within an approximation.

That it is NP-complete says that verification that a sequence indeed attains this maximum is easy (even if we don't know the maximum beforehand), but that it is unlikely we will find an efficient algorithm for the problem.

The authors note that there is a "relatively simple dynamic program solves the case of a constant-size gameboard in time polynomial in the number of pieces." So what you seem to be thinking of is solvable quite efficiently.

Bag randomizers are a modern thing, imposed by the Tetris company on all Tetris games released for the last couple decades. Classic Tetris almost universally used a standard rng, sometimes with some memory mechanism to attempt to prevent runs (ie. Remember the last piece and attempt a reroll N times to get a non duplicate piece. I believe nestris used a 1 piece memory).

You were definitely not guaranteed to get an I piece in any particular number of drops in any Tetris released in the 80s afaik. I think bag randomizers came about in Tetris worlds

The original NES variant (which is used for professional play) has random pieces.