There may be advancement for people in some nations but generally it is very limited. Loghain was born in Ferelden which does seem to allow a certain element of social mobility, provided you are not an elf. In Ferelden, an elf isn't even protected by the law; if a person kills someone defending an elf, they are guilty of a crime, even if that person is an elf acting in self defence. If you kill the Tevinter Magister in Origins you are actually guilty of this, since he is acting with the authority of the current ruler and even if he wasn't you are merely protecting elves.
It is very rare for a castles dwarf to be made a paragon; generally the dwarves are limited by their sex to the caste they are born into. That doesn't seem so different from the Qun placing you in a particular profession.
In Orlais only nobles can become Chevaliers (knights) and commoners can only attend the university with the backing of a noble, so of course the nobles can exclude them by sticking together and refusing to do so. Whilst it is possible for a commoner to be elevated to the nobility, again this is dependent on the backing of a noble and usually a bit of corruption in forging documents.
The Grey Wardens stand outside the usual rules for advancement and you have to undertake a pretty repulsive bonding ritual, after which you are doomed to haunted dreams and a bleak future; obviously a commoner would probably have an equally bleak future which is why the Grey Wardens still seem to be able to get recruits.
In between Blights, Exalted Marches and non-local warfare, the mages are pretty much limited by the confines of their Circle; it is a gilded cage but a cage nonetheless. Obviously mages have it even worse in the Qun, which is why you won't find mages jumping to join them.
So for the majority of commoners and practically all elves, the opportunities for advancement in Thedosian society are very limited, particularly if you do not have money or the backing of a noble.