#1
Do you take them on or off when you put the guitar back inside the gigbag/case?
What are the long term effects of having it on all the time :O Mines a leather strap with some fuzzy brown texture on the back.
#2
A lot depends on the finish of your guitar, chemicals in your strap, storage temp. If its a nitro finish, the chemicals may leach from the strap and damage the finish. If it gets too warm over a long period of time, the texture of the strap may imprint on the guitar. Regardless of finish and temperature, you still may get rub marks.

So... IMHO, Don't do it.
--- Joe ---
77 Bradley LPC || 07 PRS CE22 || 11 PRS MC58 Artist || 95/02 Fender Strat || 99 Gibson LP DC Std Lite
06 Ovation Elite-T || 12 Martin GPCPA4
Boss GT100 || Peavey Stereo Chorus 400 || Peavey Bandit 75 || Roland JC77