Keeping heat in house

Welcome to the forum. You could use the insulation, though most everyone on here is going to recommend a block off plate for the bottom. Basically - cutting sheet metal to fit roughly where the damper would have been. This will help keep your heat in and provide for a fire stop in case anything really bad would ever happen in the flue. Search for 'block off plate' there have to be 100's of posts on the how's and why's of using one.

An interior chimney could also benefit from a block-off plate - you can't be sure the top of the flue will remain perfectly sealed to the outside, and you probably will have some convective flow in the chimney even if that seal is good. The heat may not be lost to the outside, but it may not be going where you want it, either. Mineral-fiber insulation will still allow some air flow, so a metal plate is a good addition to that. Installers don't seem to be interested in doing them that often, so don't feel bad.
On an exterior chimney you are going to lose heat to the outdoors, not just to the upper levels of your chimney. That is especially true if your liner is not insulated and there is lots of air space. As warm air rises, the air on the exterior, cold side of the cavity will fall. As heat moves from hot to cold, the upward moving warm air wants to transfer the heat to the outside, and creates a convective loop inside the chimney. It is constantly dumping heat to the cooler, external masonry and then to the great outdoors.

If you're not a DIYer, show the how-to article to anyone who likes to play with metal and tools - they don't really need to know jack about stoves and flues as long as they know your objective.

I jsut bought some insulated blinds for my downstairs. Cost me $200 for 4 blinds. They "say" that you should have blinds in your windows for draft etc. My windows are double pane but not argon filled etc.