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.
Of course it's English. Just because it doesn't conform to the standard dialect doesn't mean it isn't English. Language is not and never has been monolithic. There are always multiple dialects, with their own histories and rules (that is, they're not some "bastardization" of the prestige dialect). It's only by convention that the standard (prestige) dialect got elevated to its position. Historically, what evolved into the standard dialect was at one point a low-prestige dialect.
The dialect you're taught in school is the standard dialect. That doesn't make it the only correct one, though. It's a matter of what's appropriate to a given situation, not a matter of right or wrong. Sure, you wouldn't wear a bikini in the boardroom, but you also wouldn't wear a three-piece suit to the beach. Language is no different.