I am going to London in February and am not sure about all the uses of the Oyster Card. Can I use the Thameslink with an Oyster Card? Even in off-peak times such as 5am or 12 am?
The rules for using the card are available fromthis page
Basically, yes you can even though you are outside the normal area for Oyster services. Make sure you do not get the Gatwick Express but, as you say in your question, get Thameslink.
Check for possible strike action and other disruption on the National Rail site here