Mephidross Vampire type changing text is a static effect and is being continuously applied. In your described case
Walking Ballista will enter the battlefield as a Vampire Artifact and will ETB with 1 additional +1/+1 counter because of
Metallic Mimic. It will gain a +1/+1 counter when it deal's damage (Not just combat damage) to a player but you have to let the trigger resolve before you can get the +1/+1 counter and if it dies before the trigger can resolve then it will not get the counter.
Example:
Walking Ballista with 1 +1/+1 counter will not be able to survive if you remove the 1 counter because State Based Actions will have it put in the graveyard for having 0 toughness.
Walking Ballista with 2 +1/+1 counters will survive as long as you do not let its toughness get below 1. In this case it would be: Remove a +1/+1 counter to deal 1 damage to your opponent, after the damage is dealt, you will get a trigger that says "Put a +1/+1 counter on
Walking Ballista" you have to allow that trigger to resolve before your
Walking Ballista will get that +1/+1 counter.