Ok, I appreciate the fact that you're willing to do some work and figure some things out, not just have everything spoonfed to you. More power too you.
1. Theoretically yes, you could improve that system to the point where you could game decently on it. However, the end result will be that you have a completely different computer, you just bought one part at a time.
2. To a certain point, yes. However, some parts need to be swapped out together or swapped out after some significant planning. Your motherboard and processor for example: which processors you can support is entirely dependent on your motherboard. If you buy a mobo that uses an older chipset, you won't be able to get any of the new processors for it and will be stuck with outdated and crappy hardware. Also, you need to plan when you buy a new case. Make sure that your motherboard will fit in it. Other than that, everything can be swapped out one at a time.
3. A good bit. Gaming rigs are known for three things: looking like a boss, running faster than anything else in the house, and costing more than said house. Ok, last is not quite true, but you get the idea. A really good rig will be worth $800-$1500 dollars. Of course, you can go beyond that, but that's overkill. But, with $400 to start with you can upgrade your GPU, your RAM, and maybe get an SSD. Save the rest, you'll need it for the CPU and mobo.
4. If you had the money all at once, it would take a week for shipping and then three days to put it all together, break it twice, and then get it working. Since you're going in increments, think a year or two. So, when you get your parts, plan ahead. Go ahead and get a really top notch graphics card now, because by the time everything else is put together it won't be quite as new.
5. Kind of, it depends on what you are working on. If you're just putting in more RAM, sure. If you're putting in a new gpu, you'll want to dedicate a weekend to that. That can often be tricky and if it goes wrong you'll have to undo everything to get your computer working again.. If you are overclocking, dedicate several weekends. That's very tricky and cannot be stopped halfway through to check your email.
6. Here's a general idea of what order you want to be upgrading things in, and what are the best manufacturers:
RAM: Corsair
Case: Corsair, Cooler Master, Thermaltake, Antec
Power supply: Corsair, Thermaltake
HDD: Western Digital, Seagate
SSD: Samsung, Crucial, Corsair
GPU: AMD, Nvidia
Monitor: Asus, Sony, AOE
Mobo: MSI, ASRock, Gigabyte, Asus
CPU: Intel
CPU cooler: Cooler Master, Thermaltake, Corsair
Oh, and one more thing. This is the build I've been working on for a while. I've been told it could play Crysis 3 on ultra. It should give you some ideas :
http://pcpartpicker.com/p/1tf3n