If I want to create a CD out of songs I'll have very slightly modified (fading, equalizing, etc.), what program should I use? I want it to be readable by CD Players, of course, and I also want it to recognize the songs individually (and thus let the user skip songs).

You could use iTunes for that.

Can I?
How would I set fades of different durations on different songs?
And how would I create a silence at the end of the songs?

Those weren't things mentioned in the OP. I'm not sure why you'd want to fade out a track (or change the duration of a fade) if it doesn't already have fade. If you're just looking to burn some CDs with crossfading or a pause in between, iTunes can do it. Otherwise Roxio Toast is a good option.

If it can help: I am currently trying to create a mix of different songs. They do not always blend well together, so I need to do some minor changes such as removing certain intros, and then fading in the abrupt start.

I'm on Windows btw.

Oh, is it Mac only? I've only ever used it on my Mac but assumed it was for both. Awesome.

EDIT:
Just looked it up, Roxio makes burning software for PC, too.

Oh, and it needs to be free. :3

I have a very resource friendly audio editor called Goldwave that's ideal for that sort of thing since it's quite powerful and fast but still easy to use. Only problem is the cracked v4 I used doesn't have a built-in mp3 encoder and thus I've replaced it with Audacity. But in your case my version would be perfect since audio CDs use the wave format so your CD burner of choice wouldn't need to convert the tracks.

For burning I recommend Ashampoo burning suite which has similar capabilities to Nero but is free. For some reason I only found a german version for free. Maybe you have more luck or you can change it to english during install.

tl;dr
Use Goldwave or Audacity to modify your tracks and any burning software to burn it.