No. If you provide the right care, your frog is very unlikely to ever have a serious illness.
Aside from the fact that males call and females don't, very little difference in general.
No. Aside from mating time, frogs are not social animals.
A healthy, well fed White's should be able to go several days without a meal. There will be differing opinions on this, but I personally would not be afraid to leave a well fed, healthy tree frog for a week without food.
The set-up costs really depend on how big and "beautiful" you want it to be. On the cheap end of things, I would say US $70 would work, but most people spend $100-$200 gettings things set-up with a nice terrarium, nice lighting, and heating.
Your month to month costs will consist of paying for the electricity associated with the terrarium (fluorescent lights don't use much electricity, nor do heat mats). The other cost will be food. Most people use crickets as food for these frogs and depending on where you source them, they can be cheap or less so. Next time you are in a pet shop, check out the price of a box of crickets to get an idea.