So far the only positive balance was found in one of the "sending addresses" in Bitcoin Core but if I understand correctly, this is not an address I own but an address I sent money to, correct?
Yes
Also, I have another wallet file that I tried to load in Bitcoin Core but was rejected. It might be a wallet of another cryptocurrency (maybe Dogecoin??). How can I check this?
There are
2 3 ways :
1. Download client for cryptocurrency you think you own/use in past and load wallet.dat into that wallet client/application
2. Open your wallet.dat with any text editor and look for text/string which represent a cryptocurrency (on Bitcoin Core, usually list of the address are on bottom of file/line).
It's painfully slow method and i don't recommend this method.For example, on my Bitcoin testnet wallet, there are 14.6K lines and i found list of my address on line 14357
3. Use pywallet (
https://github.com/jackjack-jj/pywallet) to get list of address of your wallet.dat. Check
https://bitcoin.stackexchange.com/a/25315 for more information
Make sure you backup your wallet.dat before doing anything else.
Actually just search for "name" and the address will be right next to it.