Can the task of signing a message with my address - so that I can prove I am the owner of a transaction - be done after the transaction has taken place? For example, if the recipient complains a week later that I am not the person who initiated the transaction, can I still sign a message at that point?
You can sign message from any address you own at any time and it has nothing to do with the time you made your transaction.
And It may worth mentioning that you sign the message locally on your own device without any need to the internet and without any information about the transaction history of the address.