1.) Which part do you recommend we stay in?
Near Puerta del sol if you can afford to.
2.) What do you recommend we have on our itinerary for such a short trip?
Go see the bull ring if art museums aren't your thing; it was an interesting tour.
Definitely the Retiro Park is worth a stroll through have a picnic there weather permitting.
The Spanish have some of the best parks all over Spain
Go tour the Royal Palace
3.) We leave next week, I have an autoimmune diseases so no sugar or starch intake, usually I make beef jerky and take it as a backup, how is the food? Everything pretty much pure?
The Spanish are big on meat and ham (Jamon Iberico) in general (the best is the kind where the pig was raised on a diet of olives, chestnuts or acorns) and all kinds of cured meats and cheeses. Tapas are a big part of the eating scene as well. Lunch starts around 2 and dinner around 9.
You should be able to find something with out sugar or starch at this landmark market thought it can get really crowded