Q: If Dice Override is in play, and I play a Draw Rule card which is lower than what I rolled, do I get to draw that many additional cards?

A: No. Since that card effectively turns the die you rolled to the new lower number, it is exactly as if you have replaced a higher Draw rule with a lower Draw rule. Having already drawn more than the new (lower) number, you do not draw any extra.

The same goes for Play rules. If you rolled Play All, and then played a Play 3 card, you can literally turn the die to the Play 3 side, and ask youself “How many cards have I played this turn? If 3 or more, then I’m done with my plays for this turn.”