A very familiar realm of questions. Abstract algebra often attracts, at its elementary levels, because it doesn't seem to require difficult topology and differential geometry. This is an illusion: abstract algebra is but a servant to the most important problems, which are inseparable from physics. Lie groups and algebras, by themselves, mean little, unless they are ready tools for differential-geometric problems, e.g., in mechanics and optimal control.
To see the connection, I would recommend looking at paper 18 here: http://www.math.rutgers.edu/~sussmann/currentpapers.html, and also browsing the relevant sections of V. Arnol'd's "Mathematical Methods of Classical Mechanics".
Now, to your specific questions.
This question seems to presume that
more open problems'' implieseasier to do research.'' Not necessarily the case. I think, the qualification you are looking for the presence of open problems that can be approached by first solving simpler instances, and then adding to the generality. (Which is how you should pursue any research.)
Yes, definitely. This is exactly the subject of Arnol'd's popular article "On teaching mathematics." In particular, he says that it is unreasonable to teach ring ideals to students who have never seen a hypocycloid. Although currently in the minority, he is not alone in his views: Von Neumann, Kolmogorov, and Atiyah all insisted that a mathematician should always take care to stay connected with physics and to other fields. (Unfortunately, many mathematicians today are "Bourbakists".) Pontryagin, a renowned topologist, also made fundamental contributions to optimal control theory. Also, look at the topics Von Neumann worked on throughout his lifetime. Turing, besides his work on computability, published a paper on the "Chemical Basis of Morphogenesis."
Definitely in what interests you the most (because sometimes, at peaks of exasperation, it is your ONLY remaining motivation to push through). With the following caveat: make sure you preferences are informed. (E.g., it is easy to think that something is a promising field, when in fact it is just a silo of detached and narrow problems, like "Classify all the conjugate classes of a given group.") In what you describe, you seem to shy away from differential equations, differentiable manifolds, and, generally, analysis. Not good and very dangerous, but also typical. "Algebraist" does not mean "free of analysis and differential geometry and (oh, horror!) measure theory."
One other idea is to talk to some engineering faculty who work in a mathematically heavy field of engineering (e.g., optimal control, elasticity theory, statistical signal processing). Graphs are nice, but might leave you without a background in the differentiable apparatus of mathematics.
Best of luck!