Good for you for doing some research prior to acquiring a box turtle!
Eastern Box turtles do best when kept outdoors in a climate that mimics the climate of their natural range; temperate, humid, with warm summers. In my opinion they are not suited for a life indoors year round. A well equipped indoor setup would be expensive, too. I have a 4'x8' indoor 'infirmary' set-up, and a 1000 sq. ft. outdoor pen, and the indoor set-up cost me as much as the outdoor pen did to build.
To answer your questions:
#1 They are active during the warmer part of the year, in particular the summer months when temperatures throughout their range are quite high. These higher temperatures are recommended on caresheets because eastern box turtles need them in order to digest food, etc. They also require a temperature gradient, in order to regulate their body temperatures.
#2 Most commercially available foods that I am aware of are incomplete in some way or another. Eastern box turtles eat a tremendous variety of different foods (they are opportunistic omnivores), including carrion, and commercial foods don't even come close to provideing the range of nutrients a box turtle needs.
#3 This assertion is valid. They can suffer and even die if improperly overwintered. They are very sensitive to photoperiod: shorter days are a signal to prepare for hibernation during colder weather. One of their first responses to this signal is to stop eating, in order to empty the digestive tract prior to hibernation. If a box turtle is brought indoors and is not provided with sufficient high temperatures and adequate lighting to mimic a long summer's day, it will lose weight if kept at a temperature above about 50 degrees or so, because it will be just warm enough to continue to burn calories but not warm enough to stimulate its appetite or enable to digest food.
In short, you would have to 'trick' the turtle into thinking that it is still summer by providing those conditions, or it needs to go into hibernation so that its metabolism slows to a point where it is not rapidly losing weight.
Other considerations for keeping one indoors: they require high humidity. The kind of humidity that makes us humans uncomfortable. Its what they thrive in. If they don't have it, they tend to develop repiratory problems and can become very, very sick, and yes, even die. They also do best when they have plenty of room to roam around in, which is pretty much impossible to provide indoors when you take into consideration their other needs.
If you are looking for a small turtle to keep as an indoor pet, I have heard that mud or musk turtles a better choices.
Ruby