gatwick to london

I believe Heathrow is the only London airport with direct Tube connections. You would have to start with a train from Gatwick, then connect to the Tube closer to London.

No, you cannot. Gatwick is well beyond the last zone. Unless you need to travel beyond zone 2 while you're in London, buy a 7-day, zone 1-2 travelcard at Gatwick. Take a Southern train to Victoria and buy a ticket to cover the section from Gatwick to the edge of zone 2. Your travelcard will get you the rest of the way.

The only station in zone 2 that would meet your needs if using that trick would be Clapham Junction. The train must stop there in order to legally use a connecting ticket, so don't get on a train which is first stop East Croydon. I notice that the special fares now available on Southern are the same price from Gatwick to Clapham Junction or Victoria - £5 single. EDIT:
If your zones 1-6 (there are 9 zones now - surely you don't want 1-9 as it is only valid in far northwest London suburbs) card is valid during the two journeys you could do the same thing via East Croydon which is a main interchange point and in Zone 5.

If you book a Southern train online in advance, you can get a fare as low as £3.70 for Gatwick-Victoria. However, the ticket will be for a specific departure date and time so you must be sure to allow enough time between your scheduled arrival at Gatwick and your train to Victoria. The standard anytime 2nd class fare on a Southern train is £12.70. If you don't want to worry about missing your train if your flight is late, then the "trick" I mentioned in my previous post would work well - so long as you follow Nigel's advice about picking a train that stops at Clapham Junction.

Save your inbound ticket if you are interested in the 2 for 1 offers from http://www.daysoutguide.co.uk/.

Chip is right about the 2 for 1s. Just make sure the ticket is still valid on the day you claim the 2 for 1. Most tickets are only valid for the day of travel.

Buying travelcards with the National Rail logo on them at the Gatwick train station will also qualify you for the 2-for-1 offers. (Travelcards bought from Tube stations won't.)