Game Identification – Original Version of Tetris Name

As far as I know (and I've done quite a bit of reading about Tetris lately), no official version of Tetris changes its piece-selection method as the game goes on.

Assuming you're asking about the NES version, the only difference as the levels progress is the speed that the blocks fall at.

Here is the falling speed for the levels, in the number of frames it takes for each piece to drop one cell towards the bottom. The game runs at 60 frames/second, so a value of 60 would mean it moves down one cell per second, a 6 is 10 cells/second, etc.

  Level    Drop speed
         (frames/line)
     00            48 (0.8 s)
     01            43 (0.72s)
     02            38 (0.63s) 
     03            33 (0.55s) 
     04            28 (0.47s) 
     05            23 (0.38s)
     06            18 (0.3 s) 
     07            13 (0.22s)
     08             8 (0.13s)
     09             6 (0.1 s)
  10-12             5 (0.08s) 
  13-15             4 (0.07s)
  16-18             3 (0.05s)
  19-28             2 (0.03s)
    29+             1 (0.02s)

(source)

For completeness, the randomizer always works as follows:

1. Pick a (pseudo)random number from 0-7, representing the 7 possible pieces and one "dummy number" (7). If it comes up as 7 or the same number as the previous piece, go to step 2. Otherwise, give the piece represented by the chosen number.
2. Pick a (pseudo)random number from 0-6, and give the piece represented by the number.

So there's a lower-than-normal chance of getting the same block twice in a row, but that's about it.

It is called Paper Soccer by Wikipedia, versions can be found online and there seem to be forms of it available on both Google Play and the Apple App Store.

More cultured people might call it Paper Football, but apparently that's something else.