Are there laws that specifically require that the President approve those things you mentioned - the size of the military and the pay?
How would we find out whether a law has to be presented to a valid Constitutional president before they can become law?
The size and pay goes to the President for signature as part of the Defense Budget.
“On to the President
Regardless of how it leaves the Congress, once it does, it goes to the President for his signature. Note that the legislative process does not operate in a vacuum, and the President, or his staff, has been tracking bills that pass the Congress. A bill showing up on the President’s desk, then, is never a surprise. In all likelihood, the President has commented on the bill, indicating his likelihood of signing it, perhaps indicating that he will veto it unless certain provisions are in the bill, and so on. By the time the President officially sees the bill, it is either in accordance with his wishes, or in defiance of them.
Officially, all bills that pass both houses are signed by the Speaker of the House and the President (or President Pro Tem) of the Senate before being presented to the President. This process does not usually include any politicized delays, but it could delay a bill a day or two. Then, the bill is delivered to the President and the 10-day clock starts to tick.
The President may sign the bill at any time after its deliverance. If it sits unsigned for more than the 10-day period, it becomes law regardless of his signature or not. The exception to this 10-day period is commonly called a pocket veto. In a pocket veto, the President can kill a bill if it goes unsigned and Congress adjourns prior to the 10-day time limit. The term “pocket veto” comes from the fact that if the President knows an adjournment is coming, he can place the bill in his pocket and forget about it. Note that the general interpretation of the adjournment needed for a pocket veto does not include short-term adjournments; only when the Congress adjourns “sine die,” or, basically, for good. This might be when a Congress ends before the next begins, or during an extended adjournment during a seasonal break.”
http://www.usconstitution.net/consttop_law.html