How to replace a non-standard (20cm) bathroom faucet

In the long term, I think you need to replace the faucet, but in the meantime:

You could try an epoxy with metal added (J-B Weld is the most famous, but the SuperGlue company makes a light-duty metal epoxy that I've used for a bunch of jobs; the one-use packages are available at the 99-cent store).
In either case,

• clean and dry the surfaces (turn off the water to avoid mistakes)
• mix the epoxy and apply it (making sure not to block the water flow, of course)
• CAREFULLY clean any excess from the outside surface - it will be much harder to remove once it's set
• line things up carefully, and stabilize the joint while the epoxy sets. I recommend a "dry run" - try to tape it up and see whether it will hold, or whether you'll need some sort of clamp. There's nothing worse than finding out that your joint isn't stable when you've already glued it!
• Pay attention to the directions and DON'T remove the tape (or clamp) or try to run the water until after the recommended curing time.

I don't know whether I'd trust the resulting joint to hold pressure, but since this is beyond the valve that shouldn't be a problem.

I've used channel-locks and crescent wrenches, but a basin wrench will save you some busted knuckles. I think that your 1/2" supply is probably coming out of the wall and is grey Polybutylene pipe? If so, you need a 1/2" PB x 1/4" PB shutoff valve, or an adapter to convert to PEX and then the angle shutoff valve.

If you do indeed have a 1/2" supply to your old faucet (very odd indeed -- are you sure that it isn't just the connecting nut that is 1/2"?), then you can probably replace those with braided supplies of the proper size on each end fairly easily. If you post pics of what is going on we can give you more specific advice.