Yes, a damper should be installed as close to the stove as possible in the pipe.
The stove with bent corners is a later stove built after 1980. It could be a II, III, IV or later. Pictures may be able to identify it. They were definately UL listed, and those with shield back and bottom usually had UL tag on rear shield.
There was no screen used or available for a Mama Bear. Screens were only used on double door stoves with a wider doorway to view fire. That's what makes them the "Fireplace Series". Viewing the fire with screen and open doors in fireplace mode also lowers efficiency and isn't considered a "radiant heater" as used with doors closed.
If you have a double door stove, during open door burning, the damper is closed until smoke starts to roll in at the top. Open slightly to avoid smoke spillage and maintain as much heat as possible in stove.
You will notice as soon as you light the stove, closing the damper partially slows the fire and retains more heat in the stove while getting the fire established. Closed or just cracked overnight usually works well with intakes cracked to half open or more, depending on chimney draft and heat required for overnight burn. The hole size in damper is another factor. Many newer cast dampers have smaller holes along the shaft that plug up quickly. You should have a damper with the larger metered size holes shown below.
View attachment 88537
That's nickel plating, which gets harder with heat, unlike chrome that will peel when heated.