(1) Cold-crashing won't kill the yeast. It will help drop a lot of particulate out of suspension and clear the beer.
(2) Cold-crashing will cause some amount of suck-back through the airlock, which can dump some sanitizer/airlock fluid into the beer (not a huge issue) and also suck oxygen back into the carboy (which can be problematic, and more so for hop-forward styles).
There are a lot of different ways to deal with suck-back, but most of those I would classify as advanced techniques that require a better understanding and a development of the process. I, for example, use a mylar balloon (a "balloon-lock") that fills with CO2. Easy, cheap, but still requires an active understanding of fermentaiton.
What is the style of the beer? Some styles are a lot less susceptible to oxidation problems. If you brewed a big hoppy IPA for your first beer, unfortunately, you may have bittten off more than you can chew to start. Ask me how I know
. My third beer was a Pliny the Elder clone that came off hotly alcoholic and oxidized like wet cardboard. Homebrewing is an ongoing process of learning. RDWHAHB (relax, don't worry, have a homebrew).
I would also skip the gelatin for now - its purely aesthetic. Provided your SG (specific gravity) is stable and at least 14 days have passed since you started fermentation, you should be okay to bottle.