These are more used as a means to deduce properties of the real numbers than used for their own sake. You can’t even speak of “the” hypperreal numbers, because then you’d have to specify what ultrafilter you were using. Even just proving these exist requires a form of choice. You probably don’t want to use these to represent anything.
Yes, but note that given a non-pathological ultra-filter two different versions of the hyper-reals will look for most purposes the same.
Yes, but note that given a non-pathological ultra-filter two different versions of the hyper-reals will look for most purposes the same.