What are the different fields in video game programming?
Game programming as technical discipline is a very broad area, it goes from highly abstract algorithms such as used in NPC control, to video hardware and internals of the CPU.
If you extend towards making a game from scratch, you get game design (ie what should happen in the game, what are the problems that the user solves), and graphics as well.
For indie developers, you get marketing the product as well.
Will finishing a course in computer science be enough?
No.
Education is extremely helpful in pretty much any field, and by all means do it, and try to do more.
On the other hand, education only explains how you can look at things, common methods of how to solve certain problems, underlying theory.
Like any craftsmanship, practice makes perfect. Education is useful as foundation, but in practice, problems are a tangled mess that you need to unravel before you can think about solving them. In practice, you get a 25,000 lines piece of code which "has a bug". In practice, there are version control systems to work with, issue trackers, debugger tools, and library version compatibility problems.
By encountering these things, and working with them, you sharpen your developer skills. These skills are never too sharp, they always need more practice.
Does a degree mean anything for getting into a job for game programming?
If all other things are equal, and you are the one person with an education, yes it makes a difference.
On the other hand, if another person can show he made a game, he wins, most likely.
Or do I need to focus more on learning, practicing and proving that I can make games on my own first?
I don't want to answer this one, but I'll give you a question in return:
Why (deep down from your heart), do you want to program games? I don't need the answer, but find the truly honest answer for yourself.
At a simpler level (where the honest answer may help): Why do you aim for going into game programming industry if you don't know whether you like game programming?
How much demand is there for video game programmers?
No idea, not much is my guess. Pretty much everybody can program to some extent, and sure enough, it even works for some value of "works".
From what I read online people make it sound like going into any field of game programming is like becoming a slave.
I hear things like "you'll be working 12 hour shifts 5-7 days a week or more" or "you're expendable so if you screw up more than a few times you'll get the boot". Is it really like that or is that a single persons gritty opinion?
Programming is just plain factory line production most of the time.
You're paid to convert the raw materials (game design) into a nice looking product (the code), within the deadline, and preferably as fast and cheap as possible.
Writing software is just everyday business.
I'd say if you're looking for a nice and simple 9 to 5 job, where you forget all about work the moment you walk out the door from work, programming is not for you.