Some of your questions will depend on what your aircraft selection is; I'm answering this from my perspective as an NFO (carrier jet) spouse.
(1) Are you away from home a lot during the training pipeline?
My husband and I didn't live together/near each other until after he got his wings, but I will say that training is one of the few times when you will have A LOT of time to be home. Yes, you are studying a lot. But, there is no chance of going anywhere, you're home every night (except for a cross country here and there) and you have no other job except to be a student. There were many times, weeks at a time depending on weather, etc., that my husband just hung out waiting to start the next flight phase, etc. There were also times he flew 2 times a day, studied late into the night, etc. It's not very consistent but overall, he was home a lot more than when he was in his fleet squadron when they were home.
(2) How often will I be away and for how long when I first report to my squadron?
A lot of that depends on where in the deployment schedule your squadron is when you report, and that's based on pure luck! Basically, most carrier squadrons are on an 18/6 schedule: 18 months home (really, 12 months home, then 6 months of work ups which end up being about 3 months gone in smaller blocks), 6 (or 7) months gone on deployment. For example, when my husband first got to his fleet squadron, they were in work ups so he pretty much immediately left for 6 weeks on the boat, then came home for 6 weeks, then gone to Fallon for about a month, then home for a month, then deployment for 6 months. Some guys are lucky and report just as the boat is pulling in. It depends on the needs of the squadrons as to which one you get sent to, and how soon they are leaving. A squadron that just returned home can deal with not having a full squadron of pilots/NFO's if another squadron getting ready to leave needs them. Also, keep in mind that when the squadron is "home", you will still have a ground job everyday. My husband went to work everyday from 7:30-4, plus flying if the flights were outside of these hours.
(3) How much contact does a person have with his/her spouse while deployed, whether on a ship or at an overseas base?
I can tell you that when my husband left for his deployment about 6 months post- 9/11, what was supposed to be an awesome Meditteranean cruise with lots of port calls (that I was so excited to fly out to to see him) ended up being not quite that. There were very few port calls, I only spoke to him on the phone 2 times (it was an old ship so the phones went out quite often), 1 teleconference (about 3 months into it, 10 minutes long), and daily emails.