Sony's marketing on the Xperia 1 does suggest that it is capable of shooting 10 bit video.
You'd have to do some proper image analysis to work out whether or not it really is 10 bit (counting bands like you said!) but I don't see why they would lie about it.
> does applying a gamma curve after it has already been output by the image processor actually increase dynamic range?
That depends on the exact image pipeline of the phone and camera module in question.
A few models do allow an application to define the gamma curve, but most mid-range/lower end models just bake the image to rec.709 at which point you can't really gain much if anything at all.
Filmic Pro have a whitepaper on how they manage it here though it's pretty contentious as to how effective their method actually is.
From the forum arguments I've read over it, Filmic in particular does seem to be able to get a stop or two more dynamic range on some hardware; however when you're stuck with an 8bit/4:2:0 source on most handsets you can't really take all that much advantage of it - it just falls apart in the grade.