I mean my family has lived for at least 300 years in a region close to where the neanderthals were first discovered. I assume I have some of those genes in me

Indeed, we all do! I don't think there is a single extent human population without some archaic admixture. If you're interested in further reading: https://en.wikipedia.org/wiki/Interbreeding_between_archaic_...

IIRC every human population (possibly non-African?) has some Neanderthal DNA. I believe Asian and Oceania groups also have Denisovan DNA.

All the non-African humans have a little Neanderthal DNA, and indeed some groups from East-Asia and Oceania also have a little Denisovan DNA.

Apparently East Asians have more neanderthal DNA than Europeans so I'm not sure the region of initial discovery really matter that much. Also Neanderthal 1 is 40000 years old, your family may have arrived in the region even a 1000 years ago and be completely unrelated to it.

It works the other way too though. There's a good chance that at some point along those thousand years, somebody had kids with somebody else who had older ancestry in the area, and so on.

That assumes their family only bred with each other, which I very much hope isn't the case!

Most europeans do have significant amounts of neandertal DNA, but the likelyhood of your people 300 years ago being in the same place in europe that long ago, even a thousand years ago, is slim. There’s been massive migrations and demographic shifts. The most “ancient” population in Europe I believe are thought to be the Basques.

That argument actually works both ways, as I can see certain groups clinging to the Out-of-Africa hypothesis to justify modern African immigration to European countries, and endless rounds of development aid.