How could he be trueborn? Wasn't Rhaegar married at the time to Elia Martell? Unless I'm missing something, he's a bastard unless legitimised.

Some people believe Rhaegar may have married Lyanna in secret at some point, and that based on the precedent set by Aegon the Conqueror (and possibly other Targaryens since then, I'm not sure), the polygamy would still be legitimate and children produced by the second marriage would be considered trueborn. I don't know if this is the case, though - I'm pretty sure polygamy was not common among the Targaryens like incest, and, like incest, I assume it was frowned upon when it happened even if no one actively fought it - and the fact that Targaryens are no longer (currently) sitting the Iron Throne with dragons to destroy anyone who objects means it's likely even if Targaryen polygamy had been accepted in the past people wouldn't accept Jon as legitimate even if Rhaegar and Lyanna had married.

Personally, I'm inclined to agree with you and say that Jon is a bastard either way, but there are people who believe he's actually the trueborn son of Rhaegar and Lyanna (or at least that it's a possibility).

On that note, if Jon had been legitimised by Stannis (or any other king) does the evidence suggest that it would apply regardless of his true parentage? Would Jon have been officially a Targaryen (if R+L=J)?

I have no idea. I assume if Stannis declared Jon to be Jon Stark then he'd be Jon Stark, but I don't know if that's actually how it works. It's possible if it turned out his father was actually a Targaryen then he'd suddenly become a Targaryen, or that the revelation would invalidate the former legitimization and he'd go back to being Jon Snow unless Stannis specifically declared that the legitimization stays and he is now Jon Targaryen.