Casting a spell for an alternative cost, such as a madness cost, doesn't change its mana value. For example, casting Twins of Maurer Estate (a card with mana cost {4}{B} and madness cost {2}{B}) for its madness cost will not cause Bygone Bishop's ability to trigger.

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.

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.