You can't sacrifice a Clue to pay multiple costs. For example, you can't sacrifice a Clue token to activate its own ability and also to activate Alquist Proft, Master Sleuth's ability.

Forensic Gadgeteer's last ability affects only abilities of artifacts you control on the battlefield. The costs of activated abilities of artifact cards that work in other zones, such as cycling, won't be reduced.

Some abilities trigger "whenever you sacrifice a Clue". Those abilities trigger whenever you sacrifice a Clue for any reason, not just to activate a Clue's activated ability.

Activated abilities contain a colon. They're generally written "[Cost]: [Effect]." Some keywords are activated abilities and will have colons in their reminder text. Triggered abilities (starting with "when," "whenever," or "at") are unaffected by the cost reduction ability of Forensic Gadgeteer.

Clue is an artifact type. Even though it appears on some cards with other permanent types, it's never a creature type, a land type, or anything but an artifact type.

If an effect refers to a Clue, it means any Clue artifact, not just a Clue artifact token. For example, you can sacrifice Wrench to pay for Alquist Proft, Master Sleuth's activated ability.

Some spells and abilities that investigate may require targets. If each target chosen is an illegal target as that spell or ability tries to resolve, it won't resolve. You won't create any Clue tokens.