Can academic teaching and quality research go hand-in-hand?

Sure, this is exactly what is expected of tenure track faculty at research intensive universities in the US. It is a lot easier when you have a little teaching experience and a little research experience before diving into the "deep" end, but there is no reason teaching and research cannot go together. In the US a PhD student might be expect to teach a single class in both the fall and spring semesters while a tenure track faculty might be expected to teach 3 classes each semester.

It can, and should. This does not always happen because researchers may have forgotten how hard it was to get the basics right. In a way researcers who publish things a lot should be the best explainers, in theory. In practice some fields do a lot of obfurscation in their papers so they do not know how to put explanations out simply.

In fact teaching helps you as a researcher, sometimes at the expense of students. It makes you much more aware of where your skills are lacking and where you might have hidden misunderstanding. But you can only ever appreciate this if you can accept this fact. Also thinking about the simple explanations works towards making better publications. Having a wall to bounce your ideas against surely can not be bad thing.

It isn't necessary that being good at one can infer the same in the other. I've noticed quite s few examples of this sort. But, teaching can truly attribute to research and vice versa as long as they are on the same track.