I wonder if it is possible to create (install) a Linux distribution B (e.g. Fedora) from an already existing Linux installation A (any brand or homespun essential). I could imagine people not having a CD burner or simply getting annoyed about the hundreds of CDs in their drawer because of tried-out Linux distributions.

I don't see why not the following should be possible:
- download ISO image of installation CDs
- start system A
- mount required image as a virtual CD drive
- run any installation program required from the virtual CD

I am aware that likely all distributions will continue with a bootable disk from here. But is this necessary? What is an installation program doing? Making partitions, checking for hardware, installing file systems, installing init scripts, copying software, creating a boot-floppy. I don't see why this should not be executed from running system A. Or am I just too newbie to understand?

If somebody sees a way how I could make this with Fedora (or some other interesting distibution), I'ld be happy to hear about!

It's possible. I've seen some threads here talking about this.
Vserver is one of the most interesting Linux-centered project now, and it seems to be one way to do what you want:
http://linux-vserver.org/