Can I remove the rafter ties from my hand-framed garage roof?
You will not want to remove any of the rafter ties (as you called cross brace). 

The rafter ties that span the horizontal distance from rafter tail to rafter tail at the side walls is what defines the triangle base of the rafter tie and its two mating rafters. Without that rafter tie the rafters can try to fold down and push the top of the side walls out. So do not remove these for the sake of your building integrity.

The spacing members that run parallel to and below the roof ridge line is equally important. It provides two main functions first of which is to keep the center top of the end walls straight and the front to back dimension of the building correct. Secondarily it ensures the spacing of the rafter ties and helps keep them directly under their mating pair of rafters. This can really help to keep things aligned if you ever decide to add a ceiling into the garage in the form of plywood or drywall.