Gatwick is not located within any of the London zones but you can buy a ticket to London that includes the travel card. It will be £15.80. Just make sure you ask for the 'with travel card' ticket.

When it says it is valid on National Rail it means within the zones. if you go outside the zones you need a separate ticket before you board the train to cover the rest of the trip.

Personally on the Monday I would leave a earlier that you propose just to allow some spare time in case of any delays. 10.30 from Earls court should be OK but that still depends on what time check-in closes for your airline.

For both trips a Southern Train will cost a lot less that the Gatwick Express but it only takes 5 minutes longer.