The above is correct. Counters stay on a permanent for as long as it is in the zone it was in when it got the counter. So a morphed creature keeps it counters after it morphs, etc.
205.1a Some effects set an object's card type. In such cases, the new card type(s) replaces any existing card types. Counters, effects, and damage marked on the object remain with it, even if they are meaningless to the new card type.
(this may be an old rule location, but the rule is the same).
It keeps the +1/+1 counters until it dies, gets exiled, blinks, or somehow leaves the zone of the battlefield.
Regarding the other question,
Mirage Mirror, the card, does not have eternalize. So even though it's a "copy" of a creature with eternalize while on the board, when it goes to the graveyard, it is no longer that copy and reverts back to being Mirror. This applies to any "becomes a copy of" card.
As an aside, though, Mirror "hits" the graveyard as the creature, and will trigger any abilities that trigger on death of a creature, creature type, or that specific creature.