late edit: DISCLAIMER: The pictured map is not actually a representation of the territories before colonisation. It's a hypothetical map of what countries there might have been had the continent not been colonised, thus all the names and borders are fictional and have never existed.
For good actual maps, check out native-land.ca.
Yes, sort of. Let me tell you my tale.
First, my parents nor grandparents had a significant amount of Cherokee or other American Indian items on display. It's possible my grandparents did but they had tons of knickknacks and if they did have some they weren't a significant portion. So that's a no to one of your questions.
However, I was always told I was 1/16th Cherokee. I don't remember who told me this originally. It was never a big deal, more just like a fun fact that was shared with me. I never really thought anything of it. At some point in college I remember hearing how all white people believe they're 1/16th Cherokee. For context, my mother's family that the ancestry allegedly comes from is from Dahlonega, GA which is sort of the southern most tip of the Appalachian mountains. Well, maybe not the most southern but pretty far down. So the Appalachian part tracks. Also, of course, the Cherokee were from Appalachia so that tracks as well.
So, when my grand parents died we were going through their stuff and we found some paper explaining how we were Cherokee. I don't want to oversell this paper. It wasn't a certificate. It was sharpie written on printer paper. But it was a little family tree showing how my mother and her siblings were 1/8th Cherokee. I want to be totally clear. I don't put faith in this. I don't necessarily even believe it's correct or anything. However, it was fascinating to see that my grandparents actually believed it. Moreso that I didn't make up that I was told this growing up. Does that make sense? Sort of like a reassurance that I wasn't crazy for believing I was told this. I didn't necessarily believe the accuracy of the claim, just that the claim was made.
My best guess is that this ancestor who would be my grand parent's grand parent, may have been partially Cherokee but that was exaggerated. I don't really know.
I viewed being 1/16th Cherokee growing up like I view the results of my DNA test now. Just some interesting information about my heritage but not exactly important. The way I sort of see Irish stuff now, which 23andme said I was. I don't feel some great call to return to my roots, but it's slightly more interesting than it would've been otherwise.