Chew at your risk. Any gum can be chewed with orthodontic braces, and it will very likely not break away the appliance from the tooth. The big problem is that the gum tends to distort the wires that are responsible for moving the teeth, and as such your treatment can be greatly extended or end in a compromised occlusion. Delay your gratification for the unveiling of a great smile!
None. Think about opening a tin can and the lid doesn't come all the way off. You then wiggle the lid back and forth until it snaps off. Same thing with braces.
Denture-safe gum. Gum that claims to be safe for dentures is probably your best bet but check with your dentist.