That was the issue when mine was doing it I think. Drove me nuts. I could hear them myself during the winter so I'm sure she was trying to get to them too. Thankfully they seem to have vacated for the summer.
I don't think renters insurance covers damages caused by movers from what I read (moved last year and looked into this). You can certainly save money hiring randoms from the internet, but there is risk involved due to liability like you mentioned, not to mention the possibility of theft. IMO...
I renamed mine when I got her but I doubt she really noticed. She responds when I yell "Hey!" to get her off the counter more than if I use her name anyway. I should have named her Temptations because that's what gets her attention the most!