The President does not have the right to declare war--he needs Congressional approval to declare "a state of war" or a "national emergency," both of which give him extraordinary powers. BUT: he can deploy troops and equipment anywhere in the world without calling it a "war." That's why the Korean War was called a "police action" or the "Korean conflict," and why the 1991 Gulf war was called "Operation Desert Storm." And he can keep 'em there as long as money doesn't run out. LBJ found it prudent to have the Senate pass the Tonkin Gulf Resolution, and W did likewise before invading Iraq. But those weren't requirements, and are regarded as "sense of..." resolutions.
Re. the draft: When I was draft eligible, I got the notion that the draft was unconstitutional because it violated the 13th Amendment ("involuntary servitude"). I asked a lot of lawyer and law-student friends about this. Law is a very specialized field, and none of the guys I asked was a Constitutional lawyer. But, the consensus view I got was that the Supreme Court had heard that argument and had ruled that the 13th Amendment applied to "Negro slavery," not the draft. They also cited the Constitution giving Congress the power "to raise and maintain standing armies." And drafting someone against his will didn't violate "due process" (Fifth and 14th Amendments) because the draft process had plenty of "due process" built in for a registrant to challenge or appeal local draft board decisions. What they didn't take into account was the way the government used the draft to target people they didn't like--such as Muhammed Ali and some of the better known Vietnam War protesters.