I can't imagine Arthur as a Doctor, nor an Aria.
Arias seem to be more of the analytical kind (Bon, Konekomaru) and whenever Arthur is really an idiot or is just faking it, he wouldn't fit the profile.
Same for tamer, if he really hates all demons, I can't see him using one as an aid in combat, not willingly at least, and if his hate is fake, then it would be inconsistent with his act.
Dragoon doesn't seem to fit his Templar Knight air.
I think the main reason he was made Paladin is because he is a powerhouse... and seems easy to manipulate, though, as I have said already, I don't totally buy.
As for Yuri being a Tamer, well... we don't really know how Tamer genetics works.
I could say Rin ain't a Tamer because he got more from daddy than from mummy. Then if Yukio got more from Yuri than from Satan, maybe he:
A) didn't really take the tamer test. Like Fujimoto knew tamers were rare and, if it is hereditary, it might arise questions about Yukio's origins that would compromise the twins anonymity.
B) his reject to everything related to demons made him suppress it when he took the test... now then that would be good power up for Yukio, to gain tamer talent (roots for giant demon in the calendar cover).