Thanks again to everyone for all the really great input. I'll try and get a picture posted soon to see if that helps.
We don't know for sure if the liner is insulated or not, but given some of the other shoddy repairs done by previous owners, it wouldn't surprise me.
The cast iron clean out door I mentioned is inside the house. We did reseal all the gaskets, etc, after thanksgiving, so they should be fine. It's what makes me think it's just the draft and/or needing to actually use a manometer to set the damper.
We didn't hear back from the company about coming out to reinspect, etc, yet, so we'll shut it down this weekend and troubleshoot as much as possible ourselves in the meantime til they can come out.
WNY - it does seem to be when there's no call for heat and it's just idling, and I'd totally forgotten about changing the min/max values on the coal-trol. The fire sometimes does burn back in the grate a little bit, but again, all of these problems are totally random and have no seeming pattern. Maybe once a month since we started burning. It's usually for a day or so (and we do the temporary fix of bumping up the heat and fan), and then it seems to be fine. When the company comes out, they'll actually check the draft with a manometer, as well as reinspecting, troubleshooting, etc, so we can get that set for real.
Unless we're having high winds, the damper stays pretty much closed, which is what Leisure Line said it should be doing; not flapping open and shut all the time. I rechecked it last night and it's definitely level and doesn't seem to be sticking.
Redoing the liner is not an option b/c of cost, so we'll have to see how the reinspection goes. This company sounds way better and more thorough than the last one, since they never mentioned any of these things as potential issues.
Thanks again for all the help and input!