Is the determination of which tetris block comes next random?

This is kind of a vague question and the primary answer is going to be equally vague: it depends on which Tetris you are playing.

More elaborately, every different version of Tetris will have its own block choice algorithm. The ones I've most commonly heard about are a "true" random generator that doesn't make any kind of judgement, a "fair" generator that generates random sequences of all seven blocks (so you /will/ get every type of block equally often, but the order in which you get the blocks within a 'batch' of six will vary)...

... And then there is Bastet, or "Bastard Tetris", which distinguishes itself by having an algorithm that chooses the worst possible block with malice aforethought.

For further reading I recommend the wiki pages here (different tetris random generators) and here (the 'official' guideline to what makes a Tetris game).

As I recall, the original version of Tetris randomly generated a permutation of the 7 blocks, and then sent those.

So, the first set of 7 blocks forms a full set... and the second set of 7 blocks forms a full set.

Meaning, for example, you only rarely had an opportunity to get a double.

I'd be curious to know what random generator was used. Given the time that Tetris was first released, it's likely that the randomness was quite deterministic.