The motor shaft material might not mind you welding, but of course bear in mind on the other side of the front bearing there is a lot of glue and plastic. Might be worth opening on old motor, seeing what it looks like, close it back up to test weld it, and then seeing if it looks molten or damaged on the inside. If it does, eh...

As for welding the pinion on, if you try to weld the drill shafts they become weak and brittle (something to do with the way they're made) and the gears, as I believe they're made in the same way, may suffer from this too.

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If the motor test weld goes okay, can I suggest you put a small blob of weld on the shaft, then file it down to just a fraction too big for the bore of the gear and try repressing the gear? Essentially the same thing as crimping but you're adding material instead of misshaping it. The shafts are quite tough, you can support the back of the shaft on the other side of the motor and hammer quite hard fairly safely. Just trying to avoid welding the gear as I think it might make it useless.