You can stick weld thin stuff (with a thin rod) with a single car battery. It's difficult to strike the arc and maintain a welding puddle however. 24V (two batteries) or 36V (three batteries) is the range I'd recommend playing with if you want to weld with car batteries. I've done it many times and it works reasonably well.
Car and truck alternators work well also and you can use an old interior/dash light dimmer in series with the field coil to control current somewhat. Obviously not all welding jobs demand full power and thinner stuff can blow through.
A car charger can provide it's start rating only for a few seconds, typically 15-20 max. These things are designed to suppliment a weak battery and not replace it entirely. This is why if the battery is completely dead you cannot start it with a charger/starter - the battery isn't helping at all and probably sucking more current than the stater, at least initially.
Just be very careful. Amps can kill you very easily.
