First thing, if it hasn't been done already, would be to have the chimney inspected by a certified chimney sweep. Smoke spilling out into the room usually means very poor draft caused by either blockage in the chimney, a chimney that is not high enough or a house so tightly sealed that there is insufficient air to permit the chimney to work as it should. Cold air coming down the chimney usually means the damper within the fireplace is not working correctly - stuck open, maybe even missing. Exterior chimneys are more prone to poor draft than those within the house.

Changing to a gas fired insert would indirectly solve the cold air problem. In order to make the conversion you would probably have to have an insulated stainless steel liner installed with a block off plate put in just above the junction between the stove pipe and the liner. The combination of the insulation and block off plate should end any cold air problems, but would entail considerable expense. Even a gas insert needs air to draft properly, so solving that issue would still need to be addressed even if you decide on an insert.

We have the option to use the intermittent pilot in our gas insert, but we keep the pilot lit once the temp drops. Our FP is on the ground floor of our house. When the temperature drops near (or below?) 50ð, the pilot will often "time-out" while trying to lightt. The insert manufacturer (Regency), told us we should keep the pilot lit during the colder months. An added benefit, is that it keeps the insert warm, and in so doing, also keeps the air in and around the insert (and any air coming into the room from it), a lot warmer than it otherwise would be.