No-one really knows the educational effects of reliance on SE yet, mostly because you, and your generation, are the canaries down the coal-mine. Most of the experts who answer questions on the technical SE sites are people who completed their graduate education before SE existed, and some before the Internet was even in regular use. Those of us who answer questions on these technical SE sites rely on the students who use this facility to have good judgment about when reliance on this assistance is helping them learn, as opposed to being a substitute to learning. From our perspective, we certainly hope this assistance will help your understanding of the material, and we try to frame our answers in this way, but we are relying on you to let us know how it all works out, not the other way around.

So here comes some specifics of my question:

• Is searching SE for homework problem common for Math students?

You are probably in a much better position to judge this than most of us. Ask around the students in your faculty and see if they use this resource, or if it is just you. There are certainly a reasonable number of student questions on SE.math, but the volume is still far below the number of mathematics undergraduates presently studying. This suggests that one a small proportion of math students are using SE to ask questions, though many may be looking up answers to existing questions on that facility.

• How will doing so affect a student's understanding of the material?

This really depends on the exercise of good judgment by the student, so you tell us! Has SE helped you or hindered you? Have SE answers functioned as a tool to assist your learning, or as a substitute for learning? Have there been any instances in which you used an SE answer to substitute having to learn to understand a piece of material? Was this common?

• In what ways does doing so tie to one's ability to do research?

Unless the use of SE answers has hampered your previous learning, there is no reason to think it will impose a limit on your ability to learn to do research. It takes a long time to learn to do research, which is why we have PhD programs, but if you are able to get into one of those programs, there is no inherent reason that you should lack the ability to learn the material and ultimately succeed in this area. You mention your professor's comment that you won't truly understand a subject unless you attempt to prove every theorem in the relevant textbook yourself. I would say that is rather aspirational, and it assumes that the textbook is some kind of golden-tablet that perfectly delineates all necessary knowledge in the subject. In practice, this level of engagement wouldn't necessarily occur, and it is not a necessary condition for expertise. As you acquire expertise in a subject, you are naturally going to be curious to check and re-check all the foundations, so you are probably going to find that at some point you will learn to prove all or most of the relevant theorems yourself. That might occur slowly, as you learn more about your subject and relate it to other relevant areas of mathematics.

In regard to research trainnig, it is worth noting that the major difference between undergraduate mathematics, versus research mathematics, with respect to SE, is that the latter is going to involve problems that are sufficiently difficult that it might be hard to get answers on SE. At the level of graduate research, you are expected to be developing into someone who is becoming an expert in your topic, so at that point the pool of people that can help you diminished substantially. Some researchers do ask questions on SE.math pertaining to research topics, and sometimes they get useful answers, but often the complexity/obscurity of the topic is such that it is difficult to get assistance through this medium.

• What are some possible ways to remedy this besides completely re-learning the material?

You haven't really told us exactly what deficiencies you feel you have that you're trying to remedy. Presumably you have learned something in the courses you passed, so even if there are gaps in your knowledge, plugging those gaps would require only a partial relearning of the material. Certainly, if you feel that there are areas of your mathematics education where you did not acquire the relevant knowledge, go back and have another go at them, and try out some practice problems until you gain the understanding that you missed.

• How much do I have to pay in the future for stack-exchanging through my courses?

As I said at the start, you are the canary in the coalmine here, so we are waiting to see the results, and then you can tell us.