In 2001, George W. Bush had a Republican majority in Congress, but not enough to overcome a Democratic filibuster. So, Republicans passed a broad set of tax cuts with a planned expiration date in ten years based on the premise that allowing the cuts to expire would be effectively increasing taxes. Nobody would let that happen, so the cuts would be effectively permanent anyway.
In the 2009-10 session of Congress, there was a period during which Democrats had a filibuster-proof majority, during which they could have made permanent the cuts on everyone making under $250,000, which was Obama's platform, and let the rest expire. They didn't. Why? Laziness, cowardice, stupidity, or some combination thereof. Eventually, Ted Kennedy died, and Scott Brown won the Massachusetts Senate seat, denying Democrats their filibuster-proof majority, and even then, they didn't use reconciliation to extend the under-250K earners their tax cuts so that the over-250K cuts could expire. Why? See above. Only after Republicans won a bunch of seats in the 2010 midterm did a lame duck session agree to a full extension of the cuts for another two years. Then, during the "fiscal cliff" negotiations of 2012, Obama and House Republicans agreed to make permanent the cuts for everyone under the 400K/450K threshold.
This was basically a win for the GOP because nobody who makes that much money actually does so through salary. Rich people are rich based on income that they make through capital gains, etc., which are taxed differently anyway, making that whole thing moot.
What does this have to do with the corporate tax rates? The GOP will be in a very different position. They won't be able to trap the Democrats like that. Democrats let themselves get outmaneuvered on personal income taxes because they are absolutely terrified of being seen as the party of raising taxes on the middle class. Why? This guy:
Now, point of fact: Republicans have erected an elaborate mythology around Reagan as an icon of conservative purity who would never ever ever raise taxes. In fact, he raised taxes a bunch of times, and he signed the 1986 tax reform bill, but Democrats are scared of their own shadows when it comes to the threat of being called tax-raisers.
However, corporate tax cuts? It is hard to see how Republicans play the kind of game with them that they played in 2010 and 2012 on personal income taxes. The Democratic Party right now is the party of Bernie Sanders and Elizabeth Warren and other whiners who think that corporations are EEEEEEVILLLLLL!!!!!
You don't like "corporations" because CORPORATIONS ARE EEEEEVIL???!!!!! Tough shit.
Now, what corporate tax rate follows from the fact that corporations are necessary? None. Setting corporate tax rates is a hard problem, in policy terms, as most economic problems are, and in real terms, most corporations don't pay the nominal rate because of deductions anyway. This is complicated stuff. But, the anti-corporate sentiment in the Democratic Party means that they aren't going to play ball on this the way that they did on personal rates unless there are some changes over the next 10 years.
So, who has control of what in 10 years? Um....
If the GOP has full control of everything in 10 years, they can do another extension. Or, maybe we'll have gone full-nuclear, and they can make everything permanent. (Or, maybe that nuclear thing won't be metaphorical...) But, if Democrats control anything, Republicans won't get to keep any of the corporate tax cuts they are going to put into place.
Frankly, that's a mess.