You may collect evidence 0, but you would be a lazy detective with no crispy culprits. Nevertheless, if you choose to do so, abilities that trigger "whenever you collect evidence", such as those of Surveillance Monitor and Evidence Examiner, will still trigger.

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.