Bottle carbonating after cold crashing

If it's a true pilsner, that's a pretty warm fermentation temperature. Since you used the term 'cold crash' I assume you aren't going for a standard lagering period. But none of that comes close to answering your question, does it? So when I lager or cold crash I usually let my carboy come up to room temp before going to the bottling bucket. I like to give the yeast a chance to acclimate to bottle conditioning temps before I feed them a fresh jolt of priming sugar. I let my bottles sit a room temp to carbonate...

 

No. I have done both. Letting it come up in temp to around 55ish then racked to bottling bucket and racked without it letting come up. The amount of yeast activity is so minimal and there is no yeast growth, so you don't have to worry about causing esters. I didn't have any issues with carbing my bottles when adding priming sugar at 35/40
Degrees.

 

It doesn't matter what temp you add the sugar at. Although ChuckHardSlab is correct that this is a warm temp for lager yeast, for the small amount of fermentation that occurs during bottle priming, I never worried about it and never noticed any off-flavors

 

It may get rid of some of the 'yeasty' flavors, and add some crispness. It will definitely help to clarify your beer. If you really want the yeast to flocculate to the bottom, adding gelatin during cold crashing will help even more. If you want it to be crystal clear like some commercial pilsners/kolsch styles, then adding a fining agent like irish moss to the boil, and ensuring a fast cold break by using a wort chiller, then cold crashing with gelatin before bottling, will give you just that.

Carbonation in the bottle may be a bit slower since there isn't as much yeast in suspension, but given an extra week or two, there will still be enough to get the job done.