The first of your sentences to me reads “My work drives the car”, and the third one “My work drives the car”, which I don’t think is quite what you want to say…
I don’t know the ins and outs of the grammar, but I would say “Fy swydd i yw gyrru’r car”. The only alternative to that I can think of is “Gyrru’r car yw fy swydd”, which would be emphasising the driving rather than who’s doing it.
Yes - generally, what’s either side of the ‘ydy’ in these kinds of sentences points to the same thing, so here, ‘driving the car’ and ‘my job’ are referring to the same thing. Just like ‘Deborah ydw i’ - both ‘Deborah’ and ‘i’ refer to me.