Almost time to bottle my first batch ever. A few questions.

your overall process sounds good and you should get carbonated beer with that co2 schedule. I am assuming you are going to store and serve it out of the fridge that you cold crashed in?

 

I would not put beer in the keg until I was positive it was clean and held pressure. You want to do a complete disassembly and recommend you replace all o-rings and lid gasket (unless you are positive they are new). An overnight soak in PBW followed by a good Starsan rinse should do the job.

Next you want to pressurize and let it sit overnight and check pressure. It wouldn't hurt to dunk the top of the keg in a bucket of water and look for bubbles. Much easier to fix leaks when there is no beer in the way.

You want to "burp" the keg many times after filling. Different ways to cold crash, I tend to let the keg be my cold-crash vehicle. Finally, I find it easier to just set the pressure and wait.

 

You are kegging, not bottling. No need for dextrose, though some people do carbonate with sugar in a keg. (That method is fairly rare and best for the brewers who want to be as traditional as possible. Krausening even if you like. Look it up.)

You have it sorted it appears.
Cold liquid will absorb more CO2 than warm liquid, "More" is relative. If you apply 20 psi to 38F beer more gas will go into solution (carbonation) faster than if you apply 20 psi to 72F beer. Applying a higher pressure at 72F would get you the same carbonation, eventually, as a lower pressure at 38F.

Cold crashing will help to clear your beer, known as Bright Beer. Try not to move the carboy or keg much once the beer settles or you will bring the yeast at the bottom back into the beer. You may also find some chill haze, which is another post.

Applying high pressure to cold beer for a time and then lowering to serving pressure will speed up the carbonation process, but it is not something you should try first. Too many variables, and you are probably going to over-carbonate. Just apply 12 psi for a few days, keep it at 38F and wait. 38F is your magic number too, not 42, not even 40. 38.

All beer gets better with a little bit of lagering. Not aging, just a few days or a week of quiet keeping at 38F. Like a good soup or chili it is better in a day or two after it has had time to mellow. So no need to rush things.

Since you have CO2, purge everything as best as you can. You won't be able to eliminate oxygen but it helps tremendously to transfer into a Corney keg that has been filled with CO2 a few times.
Good luck.