Is there something in a debug.log from 2011 that would tell me if there are BTC in an old wallet.dat?
What I would do in this case (the easiest approach):

Copy wallet.dat to a new directory called Bitcoin in %appdata%
Run latest Bitcoin core
Open the console and do
dumpwallet C:\Users\yourusername\Desktop\wallet.txt
Quit Core and delete your new Bitcoin directory. (Don't forget to delete it, it might become larger than 100 MB, to save space)
Save my script with Ctrl + S and name it dumpwallet.html. Open it in your browser. Copy the wallet.txt output here so that only the private keys are listed.
Import wallet to Electrum: &quot;New wallet &gt; Import private keys &gt; Copy from the webpage output&quot;

Then you can see your balance or spend from it.
The tedious approach is to use db_dump of db-util to dump all the transactions and calculate the final balance by yourself. I would not bother doing it this way.