To save trim during remodeling, use two small, flat prybars. Slide the thin end of one behind the trim, pull, then insert the second in the space that opens up. Alternate pulling each bar to inch the trim off. If finishing nails protrude from the back, use slip-jaw pliers with a rounded top jaw to pull them through.
Black stove pipe (and furnace pipe, for that matter) should be securely fastened together at each joint with no less than three sheet metal screws or pop-rivets. Stove and vent pipe should be inspected at least yearly, and replaced when signs of rusting or wear are evident.