1. I live in Phoenix. So lots of desert trails, rocks, small waterfalls, hill climbs. We also go up to Colorado so that is much more muddy and forested obviously. .
I've done quite a bit of wheeling in the AZ area...mainly from Tucson down south to the Mexican border while coyote hunting--usually in the Tumacacori Area and Santa Rita Moutains. During these trips, I'm normally in a 4x4 rental vehicle (no I don't trash them) and I'd say the
biggest limiting factor is tires. I've been in some pretty difficult areas in some sketchy situations. Highway tires are less than stellar in the AZ terrain. The best vehicle I've wheeled/hunted out of was a stock-out-of-the-box Hummer H3 with BFG AT/KO tires on it.
2. My experience is little. I've been out with the FJ 8 or 9 times in the months we've had it. Have had lots of fun but would definitely put my skill level at beginner. .
Since you're professing to be in the beginner category--where we all started--stick with a stock vehicle. A stock vehicle will force you to become a better and more proficient driver. As your skills develop, the vehicle capabilities will become the limiting factor. When the vehicle is the limiting factor, upgrade as necessary...lift, bigger tires, skids etc. Just take it easy...don't force a situation. If you don't feel comfortable, take an alternate route.
I've seen quite a few people here in CA and in AZ that have kitted out rigs, with no idea how to drive them. I've witnessed people getting stuck on an obstacle that a skilled driver would be able to make in 2WD with some spinning, or in 4WD without breaking a sweat.
I recently helped a guy and his GF (in a tricked out Nissan XTerra) get their truck out of the sand--they were buried past the hubs. I was on the same trail in 2WD. Reason they got stuck: Driver error--continued judicious application of throttle not suitable for conditions--buried them. They had no recovery gear either...no snatch strap, shovel, axe etc. Here's the kicker--they had a winch installed and didn't know how to use it!!

ound:
What I'm trying to get at here is don't think you need to have a super modified rig to go offroad in.
Some of the best drivers I've seen are guys with mildly modded rigs...small lifts, moderately sized tires (33"-35") that only use their lockers when absolutely needed.
3. My budget. Well I am not looking to spend money just to spend it but I would spend up to 4k if that is what it would get it done.
Tires and recovery gear first, then upgrade as necessary. If you can, do all the work yourself. You'll save some money and learn about your rig and how things work as you go.