is there a possibility to save every color-channel on my document as separate files (cmyk and rgb) ?
You could do it manually... Select all, then choose a channel in the Channels panel, then copy. File - New, grayscale, paste, and flatten. File - Save As, and there you go. Repeat for each channel.
If it's something you need to do a lot you could automate it with an action.
You want them to be the color of the channel, then?
Okay, so choose File - New, RGB, and make it black. It might help to have your background color black ahead of time, that way you can choose the Background preset to get the black image created all at once,.
Then before you paste, choose the channel you want to paste into, then paste. When done, only one channel will have other than black in it. When you click on the RGB "channel" (to see how your image will look in color - voila, your color.
Think of a channel as just a grayscale image.
Try Channels > Split Channels. You will get a grayscale file of each channel ( do the "split" on a copy so you will have the original to go back to in the future ). You could "Place" the .tiff in Illustrator and paint it a process color there. Then, export that individual file as a .tiff or whatever. This works on a CMYK file, but should work for an RGB file, too.
You may want to give this article a look:
Basically after using split channels, you will then convert each grayscale channel to duotone/monotone mode assigning a color to each channel.