Is it possible to replace only the axle and keep the old bottom bracket, going from cottered axle to square or octalink?
I mean, does all the types of axles have the same diameter and will all fit on the same bottom bracket (keeping in mind the lenght) or not?

Can't speak for Octa Link, but the cups are "MOST LIKELY" usable with a standard Sq. Taper.

Square taper maybe. It takes a good mechanic, who knows what to measure and has a selection of bottom bracket spindles on hand. Octalink is only available as a cartridge.

Thanks a lot! Will an axle from a 67mm bottom bracket fit on a 68mm bottom bracket shell?

There's no such thing as a "67mm bottom bracket". The shell might be 67mm wide if it started at 68mm and then was machined a little over-enthusiastically, but a good mechanic would put a 1mm spacer in there to clean up the installation.

There's a lot of "ifs" in doing your suggested swap: cup thickness, spindle track width, spindle diameter, cup hole diameter, etc. It might work, it might not. It might work initially then fail quickly due to a poor bearing alignment. Only a good mechanic who knows the ins & outs of cup & cone bottom brackets can make the determination.

I can report one success. We replaced the cottered Stronglight crank on an early 70's Raleigh with a Suguino square taper spindle and SR crank and kept the original Raleigh cups and lockring. Those Raleighs had proprietary threading and replacement cups were not easy to find. As Jeff noted, your experience may vary.

I think that campagnolo mirage bottom bracket is designed for 67mm shell

Wrong. There are two common standards: Italian threaded BB shells are 70mm width while English, French, Swiss (and others) are 68mm width. As I said, it's possible the shell was "faced" aggressively by an enthusiastic mechanic (I've done this myself).

A lot of this would be cleared up if you posted a photo of what you're working on or at least told us the brand of bike you're working on.