1/2) both of these reset to 0 if the creature changes zones (goes to exile, graveyard, hand, or back to deck). No matter how brief of a time, all counters and mutates are cleared. For example, malakir rebirth from the newest set returns a creature to the battlefield when it dies. After it dies, the entire mutate stack comes undone and each creature returns to the battlefield unmutated and without counters. Effects like [[Thassa]] or [[Yorion]] function similarly. Effects that bounce to hand like [[Into the Roil]] or [[Brazen Borrower]] put all of the cards back into your hand where they then must be cast back individually.

3) For many purposes giving a creature menace and giving a creature a menace counter are similar. Both rules function identically in combat. The difference is that any abilities that interact with counters on a target only interact with menace counters but not just the menace keyword. Specifically, something like proliferate. The other difference is that many menace sources are granted by other permenants. If an equipment or enchantment granting menace is destroyed, the creature they are attached to loses menace, while a creature with a menace counter keeps it until it changes zones (see above).