By your points:
Nuking the region would incite far more fighting than it would stop. First of all, nukes wouldn't annihilate everything in the region, but they would turn it into an infinite source of various terrorist groups. Second, some kind of conflict, maybe even World War III (this is mentioned in the movie), with Russia, China, and likes would ensue.
An airstrike on the White House was supposed to cover up all the evidence (because, evidently, in the "White House Down" universe everything is stored there and "off-site backup" is a non-existing concept). This is explained by Cale as he exposes Raphelson.
Once the nuking has failed and the crisis was averted, the second best way to restart wars is to send those troops back in. It would probably make the conflicts there stronger than they were before, because it would be seen as a new occupation and the breaking of the previous peace offer.
Of course, in a movie like this, one is not to look for too much sense, because everything is in service of humor, explosions, and witty retorts, so I'd say that there is actually an unexpectedly high amount of sense in this move.
Edit: Having just re-watched the movie, I'm not sure that Raphelson's plan was to actually start the nuclear attack. Maybe he counted on the attack to be prevented by the air strike (which would've been only seconds too late for that, had the Cale family not interfered). The possibility itself was enough to get an excuse to get war back on.