No. Patching is not done for crossed eyes, per se. It's for treating a lazy eye and forcing you to use it by blocking the other eye. Once the eye strenghthens (ie, neural connections develop), the crossed eyes can improve since now the brain is getting input from both eyes. But no, patching should not be painful.
Improve vision. In childhood strabismus, patching is used when the brain starts to ignore the vision from one eye. A patch is placed over the better eye to force the brain to pay attention to the vision from the supressed eye. This is why you can see children wearing eyepatches at times.
Usually easier to. Close the patched eye if taping on eyepads--will avoid abrasions. The goal of patching is to block incoming light into the eye, so if using an eye shield without a pad, eye doesn't have to be truly patched closed underneath.