So the only way to get the nomination to the floor for a vote is to keep debate open (with Democrats speaking) until the Democrats ask for mercy, and enough agree to vote for cloture.
No, the nomination has to be on the floor before debate in order for a Filibuster to be active. A Filibuster is the act of unlimited debate on an issue currently on the floor.
How do the Dems lose if they give up the floor? Does the nomination then go to the floor without a cloture vote?
The Dems lose because then the debate is over and the vote is taken. With 51 votes he is confirmed. A vote of cloture only closes debate early i.e. before each Senator had yielded the floor twice if they desire.
The Pubs need a quorum to keep debate going - otherwise they adjourn, right?
No, a motion for adjournment must be unanimous. The motion for a quorum vote is to keep the Democrats from going home and getting sleep.
It seems to me that this works against us - the Pubs have to be there full strength, but the Dems don't.
Incorrect, all Senators must be available during an open session unless given prior consent by the Senate to be absent. If the Democrats want to Filibuster, the Republicans can keep all the Democrats there until they give in.