How would a two-halved brain work? If it would, could we still control things like motion, and would hearing, vision, and other senses still function?
$\begingroup$ May be related: en.wikipedia.org/wiki/Dual_consciousness_%28neuroscience%29 $\endgroup$– Alex StoneSep 30, 2015 at 14:58
Basically the same as a normal two-halved brain. Circumstances in which function is altered are fairly limited. For details, see case reports on Wikipedia's split-brain page, and this episode of Scientific American Frontiers that features Mike Gazzaniga's research, as mentioned here. Gazzaniga's research was reviewed in a Nature news feature (Wolman, 2012), which you may also find helpful.
Also, yes to all questions of control, except maybe hearing...not sure how we control that normally, other than in terms of attention (see "Do we only hear what we want to hear?"), which still operates normally AFAIK. Control breaks down unusually in mostly special circumstances where stimuli are presented to only one side of the brain. Both ears usually hear most sounds, and both eyes' fields of vision overlap quite a lot, so these circumstances need to be created experimentally to demonstrate really clear effects.
I've asked this question about dominance of brain hemispheres. in the question there are examples of drawings made by split brain patients using only one hand and eye at a time. I don't know if using both eyes while drawing produces different results, but the images suggest that the ability to process memory of objects is impaired: