Can you study and contribute to physics if you are a mathematician?

Yes, technically speaking, you can cross the fence -- either direction -- between those two disciplines, but be aware of their different agendas.

Physics uses the language of mathematics to construct mathematical models of reality, but the physics is really contained in the following three non-mathematical tasks:

1. Establishing the association between mathematical symbols and reality (interpretation of the mathematical models).
2. Verifying the validity of the conceived mathematical models (physical theory verification).
3. Establishing their limits of validity.

You can be a great mathematician, but if you fail in understanding the above not-so-easy tasks, you will be a bad physicist.

Yes, if you choose your sub-specialty in mathematics wisely, you will be able to interact with physicists as much as you want and decide the amount you want to be linked between the two fields. Mathematics and physics are not in a binary from one another, but instead there is a spectrum amongst the fields.

As you note above, you see that there are links between the two, namely in GR and string theory. Let me address your two priorities separately.

1. "study these subjects in depth without much difficulties; and I'm referring to difficulties in getting enrolled in classes and being able to handle them"

Typically, getting enrolled in classes during a PhD is a formality that one may even skip and just attend the course without signing up. As for studying these subjects in depth without much difficulty, that may be harder, in particular, because the subjects themselves are, in fact, difficult and will take time and energy. And you will be more focussed on mathematical knowledge in order to work on your thesis.

2. "be able to contribute to these fields"

This is quite easy if you choose your mathematical discipline and research problems with this thought in mind. Personally, I'm solely trained in pure mathematics; however, I keep my ear to the ground to string theoretic research and go to many talks in order to glean what mathematical problems string theorists have. Much of my pure mathematical research has string theoretic implications and I am in communication constantly with string theorists as I make a concerted effort to keep that connection.

This leads to communication with physicists where I can try to help with mathematical problems that physicist contemporaries have in the context of physics research. It really depends on how much physics you learn with your pure mathematics if you learn the physics with it or if you need it to black-boxed. Many programs in pure mathematics now have Geometry and Physics seminars. Usually these programs have opportunities to focus on geometric problems that have origins in physics and might satisfy your aspirations for working in the intersection of the fields.