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.

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.

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.

An Equipment without reconfigure that's also a creature can't be attached to anything. You can activate its equip ability, but it won't become attached. For example, if you control both Red Herring and Armed with Proof, Red Herring will be an Artifact Creature – Clue Equipment Fish with "Equipped creature gets +2/+0" and equip {2}, but activating its equip ability won't result in it becoming attached to the target creature.

If you control more than one Armed with Proof, creatures equipped with Clue Equipment will get +2/+0 for each Armed with Proof you control. The equip cost is still {2}.

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.

Clues stop being Equipment if Armed with Proof leaves the battlefield. If any of them are attached to creatures at that time, they become unattached.

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.