Penalties for forgetting to draw/discard cards
Irregularities like this are rarely covered explicitly in a ruleset, so most of this comes down to "use your judgement."

For the first situation, since everyone always draws and plays 1 card, it makes the most sense to let them draw the card once they have realized their mistake. Since there's nothing more than draw 1 play 1, the only way a player would have less than 5 cards is if they forgot to draw afterward, so that would be the best way to resolve the situation in a casual game. (Of course, they don't get to "undo" any of their previous turns due to "I would have played that card instead", that would be the penalty for forgetting to draw.) Even in a competitive game, I cannot think of any reason why someone would benefit from not drawing, waiting for a while, and then claiming they had too few cards and drawing one to refill, so this would probably work then as well. 

For the second situation, either they played a card and then picked it back up by mistake, or they didn't play a card and therefore didn't take a turn. If it's the former, then they should discard the card they shouldn't have (and if it's a competitive setting get some sort of formal warning, as they should know better.) Should the card already have been played...well that's a bit of a predicament, as you can't really recover a legal game state. (Bear in mind that that since all the cards are unique, the opponents should have noticed it, so all parties are somewhat at fault.) In a super casual game, perhaps just discard a random card, and ensure this doesn't happen again. In anything even somewhat competitive, a game loss is the fairest outcome.

For the latter (drawing, not finishing the turn by playing and moving, and therefore having an extra card) in a casual environment forcing the player to discard a random card is a decent option. They've already served their penalty by losing a turn, there's no need to make it worse. It should be a random card due to the fact that the player might have gotten a possible "combo" that they wouldn't have gotten if they actually played. However, this option should not be used in a competitive game, as there are situations where a player would want to "pass" (because they like their current position,) but the rules would force them to play and do something. The safest option would be a game loss (although again, both sides are at fault for not paying attention and letting things get to this point.)