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.

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.