I'm no router expert but Ill tell you what I know:
1) The 1/4" and 1/2" sizes are the diameter of the shank of the cutter. Some routers only take a 1/4" where as some only take 1/2". Some come with interchangeable collets so you can use either. If your router only accepts one obviously you are limited to that size. If you have a choice of the two I would go for the 1/2" as there is more contact area for the collet/arbour to grip on. Naturally a larger shanked cutter will have a larger cutting diameter and as jban4us said this will allow you to make larger cuts more easily.
You need to use a bit with a smaller radius than the smallest radius on your profile but I wouldn't imagine you would have a small enough radius on your cab profile for this to be a problem. It normally applies to smaller, more intricate work, but you may find you need to use a smaller radius cutter on another routing process on your cabinet.
2) It depends on the material, material thickness, depth/width of cut, cutter diameter etc but just go with what feels right. If you have already routered one side I guess you will have a rough idea. If you are seeing dark/burnt patches on the cut edge it is because the cutting bit is dwelling too much on one spot and is creating too much heat. You need to increase the feed rate to remove the material faster which will allow the heat to be dispersed in the wood shavings. Just make sure you are not taking too much material off at once as this will also cause the bit to dwell if your router isn't designed to take that size of cut.
Routers are awesome; I doubt I'll ever make anything again without one.
Same here
