1) You need very little (by today's standards) computing power to run a machine. An old 486 will work fine. I picked up a $5 IBM ThinkPad laptop for my project.
2) The software you will run, for example TurboCNC (Shareware with a low registration cost), will simply run program code that you provide it. These instructions are called G-code, and can be generated by CAM software, or extensions in certain CAD packages.
3) Gantry style is one of several styles possible. On one, the cutter head is on a gantry, moving above a stationary piece of wood. This is called a moving gantry machine. Another possibility is to keep the cutter head bridge still, and move the wood beneath it. This is a moving table design. There are, of course, real-world strengths and tradeoffs for each design.
4) I'm not sure what you're asking. Perhaps you're asking which set of plans is good? Well, take a look around this site and notice which ones keep inspiring us. We tend to take the best ideas from each set, and make our own changes. The best way to start, is to figure out how one of these machines actually works, and then look at the plans.
5) Attachment of the spindle is fairly straightforward, depending on which one you choose. There are a plethora of design examples on this site, but the most common two are: Build a platform and bolt the router's base to it. Build a set of ring shaped "clamps" and clamp the router to the vertical slide.
Hope this gives you a good start...
-- Chuck Knight