Once you've announced that you're casting a spell, players can't take actions until you've finished doing so. Notably, opponents can't try to remove cards from your graveyard to stop you from collecting evidence.

You don't choose a target nonland card at the time Memory Vampire's last ability triggers. Rather, a second "reflexive" ability triggers when you collect evidence 9 this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

You don't have to choose any target players for Memory Vampire's last ability if you don't want to. In that case, the ability will still resolve and you'll have the opportunity to collect evidence 9.

If you can't exile enough cards to meet or exceed the required mana value, you can't choose to collect evidence at all.