do i discard on opponent's turn?

[Trinoger]

If I have more than 7 cards do I have to also discard on other player's cleanup phase or just mine

[DatAsh]

Only on your's.

[Trinoger]

Only on your's.

Do I still draw a card if I have more than seven already?

[DatAsh]

Do I still draw a card if I have more than seven already?

Of course you do, you always draw on your draw step unless a card states otherwise

[Trinoger]

Is there some maximum/upper limit to the number of cards you can hold in your hand at which point you stop drawing?

[DatAsh]

Is there some maximum/upper limit to the number of cards you can hold in your hand at which point you stop drawing?

No, there isn't. You could draw your whole deck if you want to. Seriously, you're asking pretty dumb questions... You're just making stuff up, as Arix says.