Could someone solve for your private keys by sending many transactions to the same address? I've read that it's never safe to use the same bitcoin address more than once because there is a small chance that someone could solve for your private keys (though I don't have a solid-enough background in encryption to understand all of the implications of this concept). In any case, even if you did generate a different address for every incoming transaction, what would stop an attacker from simply sending many transactions to the first address they see in an attempt to crack your private key?
So there are attacks where if you sign enough times with the same private key, the attacker, with access to a side-channel can figure out your key. 

So if you were signing with your private key a lot of times, this would make your private key slightly less secure. If you're receiving bitcoins to your address, you're not signing with your private key, so this isn't an issue.

The more important consideration here is that you lose privacy with address reuse.