Hi,
1. We've got to do something for our neighbors even though we don't like them. This means you don't like them.
2. We've got to do something for our neighbors even if we don't like them. This recognizes the possibility that you may or may not like them. Perhaps you haven't met them yet.
The way I see this is that even though should be followed by factual statements, (well, you could say 'even though we may not like them') while even if should go with non-factual ideas. So, in #1, 'we actually don't like them' (true) and in #2, 'we actually like them.' (not necessarily, see above.)
A: Is he coming to the party?
B: I don't think so.
Here, can 'I don't think so' be replaced with 'I think not'? Yes. However, I don't think so can reflect uncertainty but I think not is an emphatic negative.
Best wishes, Clive