To answer your first question, maybe, theoretically.
On the legal side of things, as President, Bush has the authority to conduct military operations in Iraq as a result of a joint resolution passed in 2002 authorizing the use of military force. Said resolution (pursuant to earlier legislation) requires the President to make a report to Congress once every 60 days concerning progress and/or events in regard to said use of force in Iraq.
My quick reading of the resolution does not however show me that there is an “expiration date.” Some legislation has “expiration dates” some does not.
For example the USAPATRIOT act specifically expires after a set number of years. Upon expiration it must be reauthorized by Congress. If the resolution for force was structured in a similar way then the Democrats could prevent reauthorization of the joint resolution on Iraq by blocking said reauthorization in the House; then the legislation would expire naturally and the President would be in legal limbo in regard to having forces in Iraq.
It’s not really clear if, while in the legal limbo, the Congress has any constitutional powers to basically order Bush to transport U.S. soldiers out of Iraq.
What control of the House does do is give the Democrats “power of the purse” so to speak. Spending bills have to originate in the House, and both houses of congress have to approve the federal budget each year (the 2007 budget has already been approved–and it had huge appropriations for Iraq IIRC.)
To legally end the authority of the President to conduct military operations in Iraq a resolution would have to pass in both houses basically revoking the earlier resolution. Even if the Democrats seize both Houses there’s not much indication such a measure would have enough support to pass. Several Democrat Congressmen still support the war and several support the idea of a timetable withdrawal not an abrupt end.
To use the power of the purse to end the war would be more likely (and Murtha has said he plans to do just that) however as I said above, not all Democrats oppose the war and the moderate position in the Dem party is a time table withdrawal, not an abrupt forced ending and cutting of funding to the military while it serves in the field.
Cutting off the “troops” would be disastrous politically and would do quite a lot to hurt the Democrats chances of doing well in 2008.
A similar incident occurred during the Presidency of Theodore Roosevelt. He planned to sail his “Great White Fleet” around the globe to show off U.S. naval power which he had worked since before his Presidency to increase. The Congress did not wish to fund it, so Roosevelt basically said something to the effect of, “I have enough money to send the fleet halfway around the world, so it is on the hands of Congress if they wish to strand the fleet,” the Congress eventually relented and approved funding for the globe-spanning publicity trip.
So the full answer is–yes, theoretically. But a serious attempt would result in governmental gridlock and multiple crises in government itself. The practical answer is no/no, there isn’t enough support for such an effort even within the Democratic party, many of the Democrats as I’ve said favor a structured withdrawal not an abrupt one.