LUKE1. I think it's moderately popular (As in it's not overused but it's also not rarely used_
2. I think
Luke can stand on it's own just fine
3.
Luke Jonathan,
Luke Martin,
Luke James,
Luke Anthony,
Luke Malcolm4. No bad associations that I can think of
LUCY KATE1. Suspect it's a bit more used than
Luke but still not overly popular
2.
Louisa is a completely different name, if you want a variant I#d choose
Lucia but I prefer
Lucy3.
Lucy Kate sounds nice together, not a fan of using a shorted version of a name as an actual name but it does sound nice and if you like it it's good!
4. No bad associations
*~Loads of Randomness~*
My PNLs:
http://www.behindthename.com/pnl/150423