I'd assume it varies between different nations. Just like in real life.
I doubt Fereldan has a formal definition of "noble", beyond the actual Banns and Arls etc. Socially, a bastard might be treated as more or less noble if they were acknowledged by their noble parent or parents. Fereldan titles don't appear to have a strict succession order, so presumably a bastard could inherit though they'd be less likely than legitimate children and relatives.
Orlais more likely does have a legal definition of noble, which bastards wouldn't meet without going through a legal process to acquire it.
I'm not sure how Altus families would feel about bastard children - probably it'd depend if and how much magical ability they showed?
Mages can't inherit titles, but at least some people consider mages to retain their noble status in the Circle - Sebastian addresses Bethany as a noble, for example. And mages like
Enchanter Illana from noble backgrounds have evidently benefited from this. Like with bastards, in practice a lot no doubt depends on the attitude and status of their parents - determined parents can no doubt ensure that their child will enjoy much of their noble status, but many will find themselves de facto disowned.
Commoners advancing to the nobility would likely again depend on the place. It seems like most places it would be possible to at least advance to the lower nobility, but probably not common and not with the full acceptance of the established nobles.
The First Warden and the Inquisitor would be more powerful than almost all nobles, but non-Trevelyan Inquisitors might not be considered entirely respectable people for social purposes.