Yep. It's a full-fledged verb
ruga. So keeping the singular "you" constant:
te rog = I ask you (I say "please")
te rogi = you ask yourself (you say "please")
te roagă = he/she asks you (he/she says "please")
te rugăm = we ask you (we say "please")
te rugaţi = y'all ask you (y'all say "please")
te roagă = they ask you (they say "please")
I have to say I have doubts that ”te rugați” is correct. I'll try to break it down, maybe I'll understand first. :) So, here comes: te rog (pe tine), te rogi (pe tine; this doesn' make much sense already, but let's see further), te roagă (pe tine), te rugăm (pe tine), te rugați pe tine (no-no-no!), te roagă (pe tine). I guess it works all the way, except for the ”tu” (pers. 2. sing.) and ”voi” (pers. 2. pl.) forms. The reason should be that ”te” in the vicinity of ”tu” is rather reflexive. ”voi te rugați” is really no Romanian. All the words are Romanian, but put together mean nothing.
The forms ”rog, rogi, roagă, rugăm, rugați, roagă” are correct, the pronoun does not work in every situation.
I think in other languages it is not really an option to do an act (?) on someone already part of a group. For instance, in German: ”wir fragen mich” or ”ihr fragt dich” does not make sense, since ”mich (ich)” is included in ”wir” and ”dich (du)” is included in ”ihr”. In English ”we ask me”...
The informal ”mă rog” means ”whatever”. :)
There is also ”a se ruga”, reflexive.