Any research on right-hand/left-hand based preferences when interacting with an interface?
There is indeed some research on handedness and user interfaces but not exactly at the level you seem to be after. Handedness matters for tablet interfaces, hand occlusion is a particular concern there.

Some references: http://hal.inria.fr/hal-00670516/en and http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.89.4546

Speculating myself a little bit, I would however be careful when drawing practical conclusions from any putative “natural” handedness-related inclination. Maybe there is such a thing and it could conceivably be measured but you need much more than that to justify any UI decision.

In particular, cultural factors (in a broad sense, including web and platform conventions) could just as well play a big role. The usability literature has always emphasized consistency and the fact that your interface is just going to be one of many interfaces with which users are regularly interacting so when in doubt always try to follow common practice, not necessarily because it is better as such but simply to avoid confusing your users. In any case, there is no reason to worry specifically about “natural” inclinations (as opposed to learned ones).