I doubt it. It took a lot for him just to go Super Saiyan. Maybe, and that's a big maybe, he'll achieve Super Saiyan 2 later in his life, but not Super Saiyan 3.

No.

Super Saiyan 2 is really just a powered up Super Saiyan, not really a proper transformation.
Super Saiyan 3 wasn't meant to happen but the foreign kids wanted more transformations so Akira obliged.
:V

Bardock never had a reason to transform any further, as by the time the Saiyans took over the planet he was probably dead.

I was wondering about the visions too. Toolo, the Kanassan that gave Bardock the ability, actually just activated some part of his nervous system. One could argue that Frieza's explosion could have put that back as normal. Who knows really? Either way, Bardock would end up alone, Berri's race killed or replaced by Tuffles and they, in turn, destroyed by Saiyans. Lovely. LOL

The whole story is a big "what-if" so who's to say it actually happened.