He is the Devil's advocate.
This is the classical expression. The term was used by the Catholic Church (from 1587 until the office was abolished, in 1983) for the canon lawyer who was supposed to argue against proposals for canonisation, i.e. adding someone to the official list of saints, the canon. The purpose of these arguments against canonisation was to test the strength of the arguments for canonisation as brought forward by God's Advocate.
Syntactically, it is ambiguous whether the modifies Devil or advocate; however, in this case it must modify Devil. That is because the Devil normally requires the definite article if you are referring to the one and only Christian Devil, which is the case here. The definite article can sometimes be left out, but that would be ellipsis; in that case, however, advocate shouldn't have an article either, because Devil requires it while advocate doesn't. Compare the following sentence:
He is Cleopatra's advocate.
He is the Queen's advocate.
Being someone's x usually doesn't require an article before x. Whose advocate is he? The Devil's! It would be odd to add the article where it is normally left out (with advocate) while omitting it where it is normally used (in the Devil).
He is a devil's advocate.
The indefinite article sounds less idiomatic. The article the as above could be left out in casual use; but then it would sound odd to use a phrase almost identical to the full classical expression the Devil's advocate, having merely swapped one article for the other. If you mean to say that a specific person answers to this description, use the; if you were mentioning the general concept of being a devil's advocate, you could very well use a.
He is playing devil's advocate.
Here the article is dropped in a casual manner, and the phrase is used loosely in a slightly changed environment: this is how the phrase is most often used.
You could say you are doing something for the sake of argument.
But, really, "devil's advocate" is a familiar phrase that few people take exception to. In fact, it's even used by religious people. For example,
During the canonization process of the Roman Catholic Church, the Promoter of the Faith (Latin: promotor fidei), popularly known as the Devil's advocate (Latin: advocatus diaboli), was a canon lawyer appointed by Church authorities to argue against the canonization of the candidate.
If those religious worthies may use the term, I don't see anything preventing you from doing likewise.
Best Answer
In current common usage, "playing devil's advocate" is arguing a position you do not genuinely support. It is not necessarily arguing 'for' or 'against' a proposal - it could be either. So, I think either situation you suggest would fit. (If you genuinely support the proposal/position/idea, then you're simply agreeing, or supporting, or as Jack suggests advocating.)
If the proposal is "Let's move to a new house":
"I don't think we need a new house, but to play Devil's Advocate ... "
"I agree we should get a new house, but to play Devil's Advocate ..."
Either of these looks fine to me. "For the sake of argument" is used similarly.
(jwpat7 has a more interesting answer, but I don't think the terms/offices he suggests would be recognized in the same (idiomatic, common) contexts as D.A.)