If you are casting a spell from your graveyard (for example, a spell with flashback) you can't also exile that card to pay the alternative collect evidence cost offered by Conspiracy Unraveler.

If you cast a spell for another cost "rather than pay its mana cost", you can't choose to cast it for any alternative costs. You can, however, pay additional costs. If the spell has any mandatory additional costs, such as that of Demand Answers, those must be paid to cast the card.

If you cast a spell with the alternative collect evidence cost offered by Conspiracy Unraveler and you also collect evidence as an additional cost to cast that spell, abilities of permanents that trigger "whenever you collect evidence" will trigger twice. This is also true for abilities that trigger "whenever one or more cards leave your graveyard." For example, let's say you control Conspiracy Unraveler and Evidence Examiner. You cast Crimestopper Sprite and choose to collect evidence 10 rather than pay its mana cost. You also choose to collect evidence 6 as an additional cost. Each of these will cause Evidence Examiner's last ability to trigger, and you'll end up investigating twice.

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

If a spell has in its mana cost, you must choose 0 as the value of X when casting it without paying its mana cost.

If a spell has an additional cost that includes collecting evidence, such as that of Analyze the Pollen, any additional effects that occur "if evidence was collected" will occur only if that additional cost was paid. Using the alternative cost from Conspiracy Unraveler will not cause those additional effects to occur.

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.