How long do I need to wear braces for a bilateral cross bite on my top teeth?

12 to 18 months. Probably 12 to 18 months depending on the severity.
Dental or skeletal? Braces fix dental crossbite. Skeletal crossbite correction in adults often requires surgery coupled with expansion appliance. Time to rx depends on the nature of your problem. Go see a qualified orthodontist for an initial diagnostic examination and discussion of your treatment opinions.
Perhaps tough. Bilateral skeletal crossbites in adults may require what is called a surgically-assisted rapid palatal expander. Expansion of the upper jaw for an adult needs special treatment. An opinion with an orthodontic specialist will let you know what is required and time involved.
It all depends. On the severity and extent. Best answered by the treating orthodontist. Teeth cannot be rapidly moved as that can have other negative effects dentally.
It depends. It varies based on the situation. Impossible to say without a thorough exam.
Unknown. As you are not growing, your crossbite correction will either be dental expansion (only if mild:mild) otherwise surgery is indicated. By the way you say top teeth, by definition, a crossbite is the relation of upper to lower teeth and both will be involved. Advising time on the net is useless. I hope your case is mild and best of luck!
Depends. Widening the palate can take several months especially at age 36. It may not even be possible without surgery. Orthognathic surgery or rapid palatal expansion may be required.