May? They have absolute power of the purse, they just don't have a leadership that will use it. Ask Michele Bachmann what happens when you try to defund Boehner's funding of ObamaCare, settled law don'tcha know.
Yes and No.
Congress can not unilaterally change existing law without POTUS signature. Obamacare funding is already passed by congress and signed into law by Obama. Congress has no veto powers over existing laws.
Congress can only control future spending, because spending bills originate in the House.