First time kegging tomorrow, questions

1. CO2 dissolves better at lower temperatures. I carb mine at whatever temp I have my kegerator set which is usually 40F.
2. No, I always cool it in the keg when it is ready to be carbed. Yes, flush the head space twice.
3. Depends on how quick you want it to carb. I usually just set mine at 35psi for a day or so and then dial it back to whatever pressure I'm going to be serving it at. It is usually ready in about a week.

 

You can transfer to the keg at any temp. But to carbonate, it takes less PSIs to reach a given volumes of carbonation if the temp is lower.

No problem if you cool after transfer, but either way, you should purge the keg with CO2 before and after transfer.

I recommend the set and forget method... cool your beer down to serving temp, and use the appropriate PSI setting to reach the desired CO2 volumes at equilibrium. It will take a couple weeks to carbonate, but it's foolproof.

Edit: Also, do yourself a favor and download "A Bottler's Guide to Kegging" from the page at this link...
http://www.homebrewersassociation.org/pages/zymurgy/free-downloads

Also, "From Flat to Foam" is a good resource...
http://www.homebrewersassociation.org/attachments/0000/4954/From_Flat_To_Foam-Ray_Daniels.pdf

 

1. Chill it, if you like. Or not. It'll carb just as quickly and easily either way, provided you set the proper pressure for the temperature.
2. Chill it wherever it's convenient. It doesn't matter. Though you'll probably find it easier to chill it in the keg rather in the fermenter. Definitely purge the headspace with CO2. This is not optional!
3. Neither. Set it to serving pressure in the cooler. Come back in a week and a half or so and drink perfectly carbed beer. No surprises.

 

I crash cool my beer before I keg it down to serving temperature. Definitely purge your kegs with CO2. I tend to attach the CO2 line to the "out" valve and shake it up at a pressure of 30psi for a minute or two to give it a head start on carbonating. It really does take about a week at a "normal" psi to get to a point where you like it. Here are 2 nifty tricks that I recommend you try:
Buy a spare set of poppet valve disconnects and transfer your beer IN to your keg through the "out" valve and attach one of the disconnects to the "in" valve to relieve pressure. There's less exposure to O2 this way. IF YOU DO THIS I HIGHLY RECOMMEND you find out how much a full corny keg (I'm assuming you're using a corny keg) weighs when it's full of liquid. Transfer while the keg is on a scale and you're life will be a lot easier.
Have fun!

 

What mechanism is pushing the beer into the keg here?

 

Sorry, i should have explained that. I use an auto siphon. Auto siphon connected to a tube that connects to the poppet disconnect