Logs from Bitcoin Core activity I can see from the date of the file (2014) that Bitcoin Core has had activity on my computer, but I am sure that I was not involved in Bitcoin whatsoever during that time. However I was in the early days. Could I have had a wallet that some hacker have emptied? This is from the debug.log (I can upload the whole file if necessary): 

Committing 24771 changed transactions to coin database

ProcessBlock: ACCEPTED
2014-02-07 15:12:11 stored orphan tx 96ab951dca26ccd71......(lot of nubers)

Added 1 addresses from 60.242.28.141: 11 tried, 11727 new

received block 00000000000004dd9f5923384efd1440b.... etc....


Is this some proof that actual Bitcoin transactions of coin have been taking place, or just normal startup log text?
No, that text is not proof that there were transactions from your wallet. It just means the the software was running and those lines are normal. They just indicate that the received data from peers, but that data is not necessarily related to you.

To check if you have any Bitcoin, you can run the software and see what it tells you. It sounds like you are using an old version of Bitcoin Core, so you should probably upgrade to the latest Bitcoin Core first as it will sync much faster. Because that is Bitcoin Core, you will need to download the entire blockchain which is 180+ GB of data, but you can enable pruning to reduce the disk space used.