Student or Learner
who are you playing with?
who with are you playing?
with who are you playing?
Are they all equally correct?
(1) Your first one is "good" English now accepted by everyone.
(a) Of course, the "perfect" answer is "WHOM are you playing with?"
(i) The rule is very easy: ALWAYS (no exceptions) use the object form after a preposition.
(a) If you change the question to "regular" (non- question order), you can see this more clearly: You are playing WITH (preposition) + WHOM (objective form).
(2) Your second sentence is 100% UNacceptable. Even uneducated native speakers would NEVER use that order.
(3)Your third sentence would probably never be said. If a native speaker wanted to use that word order, s/he would probably "sense" that WHOM is necessary (even if s/he did not understand the rule): WITH WHOM are you playing? To whom are you speaking?
(4) In summary:
(a) Most people say: Who are you playing with? Who are you talking to?
(b) They should say" Whom are you playing with? But it is too difficult to pronounce "whom."
(c) A few people say, "With whom are you playing?" Very elegant English.
For example: Excuse me. Would you please tell me with whom you live?
(Regular English: Hey! Who do you live with?)
(d) Nowadays "whom" has almost disappeared from conversational "American" (the English language as spoken here in the States).
(i) About the only time native speakers feel the need to use it is in the phrase "To whom it may concern."
(e) Of course, educated people try to use it in written work. But even there it is sometimes not used.
Have a nice day!
With all that said and well understood I would like to add something I have heard.
It seems old grammarians used to prescribe it is not correct to end a sentence with a preposition (based on an analogy with Latin). I have heard people used to be taught that "rule" in American schools (maybe TheParser could confirm it or not), but nowadays such "rule" has fallen.
Personally I find it a good exercise whenever I find a sentence which ends in a preposition to try to rewrite it in an equivalent way, circumventing the "problem."
Not a native speaker
"Who are you playing with?" is definitely the most commonly used way of saying it. "With whom are you playing?" is the absolutely "correct" way to say it.
***** NOT A TEACHER *****
(1) You probably know this story, but just in case that you do not, .....
(2) Supposedly ( and I emphasize supposedly), Sir Winston Churchill,
the famous prime minister of the United Kingdom, was once angry
because some government official had changed one of his sentences
that ended with a preposition.
(3) Sir Winston supposedly wrote back:
The rule which forbids ending a sentence with a preposition is the kind of nonsense up with which I will not put.
(i) To a native speaker, that sentence is very unnatural.
(ii) The natural form would be something like: The rule ...the kind of nonsense which I will not put up with."
(iii) Sir Winston was mocking those who say that ending a sentence with a preposition is "bad" English.
(4) By the way:
(a) There are many different versions of what he supposedly said. One version involves a naughty word used in the U.K.
(b) If you wish to read wonderful English writing that is clear and vigorous, please do read some of Sir Winston's many books.
Have a nice day!