Best immediately — crisp gnocchi softens within about 20 minutes. Leftovers keep 2 days and revive best in a hot dry pan rather than a microwave. Don't freeze cooked gnocchi; the texture collapses.
Correct, for shelf-stable packet gnocchi. They hold enough moisture to cook through in the pan in 6–8 minutes, and the result is better than boiling.
Usually but not always. Parmesan is traditionally made with animal rennet, so strict vegetarians should look for a labelled vegetarian pesto.
Overheated, or the olive oil was blitzed too hard — both cause bitterness. Add it off the heat and stir gently.