You don't choose a target for Sample Collector's ability at the time it triggers. Rather, a second "reflexive" ability triggers when you collect evidence 3 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.

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

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.