I’ll be speaking out of turn here given my inexperience. But hopefully this may be a boon rather than a deficit.
Are you familiar with Occam’s Razor?
In short, the simplest reason is often (but not always) the correct one. In this case, let’s focus on your first paragraph.
I failed my oral exam unexpectedly. My advisor had the impression that it would be less formal than it was (or at least that is what he told me) and I wasn't prepared. I thought it would be more along the lines of a project proposal, but instead my committee wanted my thesis to be close to half done.
Before going into the first exam, how did you prepare? What information led you to presume that the exam was just a proposal? Analyze your information sources and deduce what is missing. If the missing link is input from the committee members, then it would be wise to reach out to them and cover your bases and get their opinion.
It is not. The problem I have been working on is fairly unique, and there wasn't a lot of existing work for me to base it on. I've needed to run a lot of numerical tests to ensure my conjectures are correct before trying to prove them. Consequently, progress has been slow. But there have been some results, and I know that it is a good idea.
My second oral exam will be at the end of spring, and if I do not pass then, I will be removed from the program. I'm sure I could finish this in time to graduate, but with a hard, career-endangering deadline in sight, I'm becoming very worried about the timeline. I didn't expect to be in this position.
I've contemplated switching problems to something with a more
predictable timeline, but at this point, starting over seems like a
poor option, even if the new problem would be faster. I've thought the
same about my advisor. It seems like time has locked me into playing
what I thought was a good hand, but am now realizing may not be.
You have a thesis plan. You have a timeline. But I ask, given the restrictions, can you reasonably modify it to fit the new deadline? If not, what options would you have? Defend an incomplete thesis? Continue and complete your research as a post doctorate? Request an extension?
Are there any suggestions or advice for how to pull research together quickly for a deadline? (Other than work like hell, which I am doing.)
I would urge caution and a different approach. Working like hell is good, but burning yourself out will not help you perform well in the long run.
Work smart as well as hard. Contact your committee members for their input as well as your advisor. If you have other commitments (like work, volunteering, etc) reconsider whether or not it would be possible to keep your commitments moving forward.
Lastly, remember to leave time for yourself. No one operates on 100% at each moment of the day. Remember to eat, exercise, and find time to socialize. Practice a holistic approach to keep you at 100%.