Post Undergraduate Research I am currently an undergraduate student who has a strong interest in becoming a theoretical physicist. I have applied to 4 physics grad schools this year but it seems that my chances for going to grad school this fall 2021 is dismal; I got rejected from my safety school and a top physics grad school and the remaining ones are also quite tough to get into. I do not think I will be able to make it to the other grad schools. As of now, I have done well in undergrad but my major flaw is that I had zero undergrad physics research experience (I have done two self studies, one a graduate Quantum Mechanics course and another a Mathematical Analysis course, but that's about it). I was about to get a research experience (voluneer) during the summer 2020 but due to the COVID 19 pandemic that got cancelled. Basically, I didn't want to multitask during my undergrad years with courses + research because I didn't want to mess anything up.
Here is what I want to do in graduate school: I want to go into Mathematical Physics, with emphasis on String Theory Methods. My long term goal is to study the very early universe where quantum gravity is important and where all 4 fundamental forces are thought to be unified. For grad school, however, I want to focus on something more concrete, something people see (or can imagine) in their every day lives and so my focus for grad school is string theory methods for condensed matter physics.
What I Did during Undergrad: I am a Physics and Mathematics major (Bachelors of Science with Highest Honors) and I plan to graduate this May 2021 (since I will be finished with my requirements I will have to graduate this May as per my university policy). I have taken the Physics GRE during my junior year.
My Questions
(1) Should I work for a professor in experiment(from my school) (particle physics or condensed matter) from May 2021 - August 2022? I am willing to volunteer with no pay.
(2) Should I work for a professor in theory (from my school) from May 2021 - August 2022?
(3) Should I ask a mathematics professor (from my school) for informal mentorship from May 2021 - August 2022?
(4) Can I apply for an REU or is that too late?
(1) Should I work for a professor in experiment(from my school) (particle physics or condensed matter) from May 2021 - August 2022? I am willing to volunteer with no pay.


(2) Should I work for a professor in theory (from my school) from May 2021 - August 2022?


(3) Should I ask a mathematics professor (from my school) for informal mentorship from May 2021 - August 2022?

It may not matter whether you want to volunteer without pay or not, for two reasons.
First, it may be illegal or at least against policy. There are often laws and rules to protect workers which may include a minimum wage. Students are sometimes an exception, but you are not a student. There are fields where there are job openings and funds to pay post-graduate employees, for example this is common in my own field of neuroscience and in biology more generally (though often people hired for these sorts of jobs already worked as a student in the places they are hired); I do not think the sort of physics you are talking about is one of those, but it's also not my area and others may know better.
Second, mentorship of someone at the undergrad level is a lot of work with little concrete reward. Many professors are willing to put this time and effort in, but they may feel a responsibility to reserve those efforts for students at their institution. It may be difficult to justify spending time on someone not a student when many students are looking for a spot.
I think your opportunities, if they exist, will be rare enough that you need to consider all of them and not narrow it down to one of these three categories.

(4) Can I apply for an REU or is that too late?

I don't know much about REUs but as far as what I do know they require you to be a student pursuing a degree. If you aren't a student anymore you are probably out of luck. They are also highly competitive from what I know.

You want to do research in a highly competitive field without a lot of financial backing. You have very specific and narrow plans, which might help with your motivation but may also close a lot of opportunities for you. I suspect your best bet will be to pursue a masters degree and get some research experience in the process, but this may be expensive. I think you've made an error by not pursuing research earlier if research is what you wanted to do; gathering some experience in research and getting good recommendation letters from advisors would be much more valuable in a grad school application than improved grades in courses by avoiding research entirely. Unfortunately, not much to do about what's in the past.