Is it an issue if I've been driving the car for a few months before doing this process?
No, it can be done at any time.
Does it matter what the original color of the vehicle is?
Yes, and no.. If you pay a bit more to have all the door jams done, then the original color won't show at all. If you skip the door jams, of course it will show when you open the doors. But in my case (see below), the original color was dark blue, and it really doesn't stand out that it's different from the Autoflex Deep Sea Blue. But if my car was originally white or red, it would look pretty bad.
edit: If say your car is multi-coat red and you do the burgundy color scheme, it would probably look OK.