During the first main phase: I have a 1/2 Thraben Inspector out. My opponent casts Harnessed Lightning targeting the inspector to kill it. I then tap, in response, the Inspector to crew my Smuggler's Copter. Is this legal? Is the Copter crewed and ready to attack in the combat phase?
The Crew ability doesn't have any timing restrictions on it, so it can be activated any time you have priority. And once the ability has been activated it doesn't matter what happens to the creature(s) you tapped to activate the ability (even if the Crew ability is still on the stack).
Tapping creatures is a cost of the crew ability. And there is no point between when you have announced the crew ability, when you payed the cost of tapping a creature, and when the ability is put on the stack for your opponent to respond. Once the ability is on the stack removing the crewing creature will not do anything because the cost of activating the ability has already been paid.
From the Gatherer Rulings on the Copter:
Once a player announces that he or she is activating a crew ability, no player may take other actions until the ability’s been paid for. Notably, players can’t try to stop the ability by changing a creature’s power or by removing or tapping a creature.
Creatures that crew a Vehicle aren’t attached to it or related in any other way. Effects that affect the Vehicle, such as by destroying it or giving it a +1/+1 counter, don’t affect the creatures that crewed it and vice versa.