Interior painting during a cold winter is very doable. Some problems you will run into include slower drying and curing of the paint. Fumes could also be another problem if you use oil base paint. Most if not all regular acrylic house paints are low VOC now so fumes is much less of a problem. If you are sensitive to paint fumes I recommend using a no-VOC paint.
Other problems with painting during cold weather are;
1- Condensation forming on windows. This is one good reason to ventilate during and shortly after painting.
2- The interior temperature could slow down the drying time a bit. Modern acrylic house paints dry very well in cool temperatures. This could be waiting overnight to apply a second coat. To help you could use a fan to circulate the air in a room.
Overall there isn't many problems with interior painting during the cold winter months. The best practices are paying attention to interior humidity and apply thinner coats to help with drying.