No. Keep in mind that you may only cast spells when you have priority. If Mind Rot is cast targeting you, you can cast an instant in response while Mind Rot is still on the stack, but you will still have to discard 2 cards (or your entire hand if you don't have that many) when it does finally resolve. No player receives priority while a spell is resolving, so you cannot cast a spell that you are discarding (unless another effect would explicitly allow you to, like the Madness ability).

[[Grave Scrabbler]]

Madness is a replacement ability, meaning that it replaces the event of discarding the card with playing it. that's why you don't have to discard anothr card: it is discarded, but the discard is transformed

You can only play something like Path to Exile before the discarding happens. You can not cast a spell or use an activated ability during the resolution(When you would do the actions printed on the card) of a spell or ability that would you discard. Cards with Madness is the of the few exceptions to this rule along with cards that abilities that can trigger when the card is discarded, but even those abilities don't get to be used until the spell is resolved.

That depends actually. If you have only two cards in hand, you can play [[[path to exile]]] before [[[mind rot]]] resolves and discard only one.

608.3. If an effect attempts to do something impossible, it does only as much as possible. Example: If a player is holding only one card, an effect that reads “Discard two cards” causes him or her to discard only that card. If an effect moves cards out of the library (as opposed to drawing), it moves as many as possible.