I'm also a software engineer, and have been for seven years. For what it's worth, I was the same way after undergrad, for about a year. I'd get home from work exhausted, and feel like I had no free time, and not feel like doing anything, and just watch TV or play video games. For about a year. After 12–18 months, I got used to it. Now I feel like I do have free time. I get off work, and I code for fun. I work on personal projects. I work on hubski now. I take tai chi twice a week. I went to grad school full time while working. And about 60% of the time, my work is boring. That's why I code for fun. I can't say you'll be the same, but, that was my experience.
I have to ask the same question. I've been working for three years full time and I'm still exhausted when I get home - and didn't even do college.
I think it helps doing work that uses your skills. I'd probably still be that way if I were still sweeping floors. Does your work use your skills? if not, could you find work that does? Or is there a skill you'd like that wouldn't be too hard to get, e.g. at a trade school, e.g. welding?
First question: not really, not always, and never all of them. Second question: No. I'm already lucky enough to have found this job (which as far as job goes it could be MUCH worse) - but I only have high school education, formally, and further education is out of the question (and not even guaranteed to land me a job that would use enough of my skills to satisfy me - I live in a large city with a last reported unemployment rate of 8%, and most jobs that I would very much enjoy would require college or university studies because of the vast amounts of education facilities here). And all skills are very hard to get for me because of my situation - I have pretty much nil spare time or money.