So I was wondering what the current hot topics of E&M are. What are the unanswered questions? What kind of things would I be studying/researching if I followed the E&M path?
I do not know if you can study something like pure E&M path at a physics study programme. It is more likely you would be able to do that in an engineering study programme, e.g. electrical engineering.
It is hard to give a complex view of current research, so just a few incoherent thoughts that come to mind :
Unanswered questions? Many, but it is best you find your own during your study. Sometimes the best question to study is about something everybody thinks is known perfectly and does not need further study.
Current topics for research?
On a more engineering side, stuff like design of microwave antennae (cellphones), electron microscopes, particle accelerators - acceleration cavities use EM theory much.
I physics, large field is the behaviour of plasma, in laboratory (research on controlled fusion in hot plasma - how to make fusion work at a power plant ?), in atmosphere and outer space (physics of atmosphere - how is it possible the corona is so hot? solar physics, solar wind, etc). There is much EM in geophysics too - how does Earth's magnetic field come about ?
The above do not use quantum notions much; largely based on kinetic theory, hydrodynamics and EM.
More towards theoretical physics, there is the research on superconductors, where magnetic fields play great role; much quantum theory of fields, and computer computations (it is said nobody knows how modern superconductors work at those high temperatures).
On a very theoretical side, there is has been (very interesting I think) effort of some theorists in the past decades to extend classical EM theory to microscopic domain and throw light on phenomena like thermal radiation spectrum, van der Waals forces & Casimir forces and stability of atoms, with help of stochastic methods (stochastic electrodynamics) as an alternative to standard quantum theory. This had some successes, but still has many limitations and open questions and thus is still rather minor effort on the fringes of research.