Cold crashing while dry hopping will not hurt the beer.
Dry hopping at room temp versus serving temp is a discussion with proponents on each side.
My current process is to dry hop in the keg at serving temp while carbing.
I like the results for two reasons, one, I can't tell the difference in taste from when I used to dry hop in secondary and two, it's one less step in the process.
The length of time you dry hop is another personal preference thing. Leave the dry hops in for a few days and taste the beer, wait a few more days and taste again, rack the beer when it tastes right to you.
I leave the dry hops in the keg for the duration with no apparent, to me, negative effect.