Yeah, France is a little bit out for me. There's quite a big pond between here and there. :)
So... joysticks, and buttons. Don't those have microswitches? With connectors on each switch?
Crimping the correct mating terminals can be more reliable than soldering. (Crimp-on connectors are what keeps airplanes up in the sky.)
Perhaps the right answer to soldering arcade controls is to stop soldering altogether. :)
Solder does have a big advantage, though, in that it is very universal. Any random copper wire can be glued onto any random terminal with molten metal, and that's pretty neat because it doesn't take any specialized parts or tools to make this work.
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Anyway, I promised to help. For that kind of soldering, my usual workflow goes like something like this:
1. Cut off the end of the wire (for nice clean copper) and strip back the insulation a bit.
2. Heat up the iron and prep it (apply solder, wipe it off along with oxides, maybe apply more solder; it should be shiny and wet, but not with globs of stuff on it)
3. Tin the exposed copper wire. Heat it up while applying solder, until the solder flows through the strands freely. Too hot/too long means that a bunch of solder flows down past the insulation, which isn't ideal. Too cold/not long enough means that it kind of globs instead of flows. Perfect is somewhere in the middle. Set it aside to cool down.
4. Tin the terminal. Just heat it up with solder applied, until there's a neat little easily-flowing pool of it, and then let it cool down some. If the solder's flux core starts to burn, you're taking too long. If it blobs, it's not hot enough or not clean enough. Somewhere in the middle is, again, perfect.
5. Stick them together. Place the tinned wire on the tinned terminal so they "want" to rest together naturally (there's jigs that help with this kind of fixturing). And then, heat up the combination of the two with the iron. The solder that's mixed up with the wire will become one with the pool of solder on the terminal, and that's good. Often, there's no additional solder required for this step (but sometimes it's useful to add more). Again: Too hot and the solder flows away and the flux burns. Too cold, and it looks weird and instead of glassy. In the middle is good.
6. Remove heat. Don't move anything but the iron; just let the temperature drop until things turn solid.
Anyway, wire is easy to find. And a bag of cheap switches shouldn't cost too much. :)
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Now, that said: I live in the States, where we never adopted RoHS. It's very simple for me to get solder with lead in it and that's the usual thing to see on any repair bench.
Lead-free solder is relatively unusual, and is not something I care to work with any more than I have to. I do not enjoy encountering it during repairs, mostly because it seems like it takes an astounding amount of heat to get it to do anything but laugh at me.
And really, for occasional hobbyist use, lead isn't a problem: Wash your hands afterwards, keep your work area clean, and don't let kids play with it. :)
What kind of solder are you using?