1 & 2 You could get the train from Gatwick to Clapham Junction which is one stop from Wandsworth Town or a short walk. Tip - download CityMapper for your phone
https://citymapper.com/london. It integrates all transport - tube/bus/rail/walk/Uber - to give you the best routes anywhere in and around London. It tells you when to get on and off and which carriage is best.
3. It's not in the Centre but there are good transport links. Tubes and buses run all night and it would be worth downloading Uber if you don't have it. Use Citymapper to find bus routes into the centre especially as you're staying south of the river. One of the pleasures of London is sitting on the top deck going across one of the bridges, especially in the evening.
4. What sort of thing do you like? Tourist traps are shite and there are great places to eat all over the city. Don't go to the touristy bit of Covent Garden but look out for pre-theatre menus which are excellent value. Loch Fyne is a great fish restaurant - pricey but the pre theatre menu is not too bad.
Try somewhere like The Merchant's Tavern near Old Street for a special night out
http://www.merchantstavern.co.uk/, Angela Hartnett's Restaurant. It has a great DJ playing vinyl/ cocktails and fantastic food. get there on the magic orange train (Overground) direct from Clapham Junction to Shoreditch High Street. Loads of new and interesting restaurants round there. And don't go to Brick Lane for a curry - it's a tourist trap and horrible. Try Dishoom
http://www.dishoom.com/ there are a few branches (I like the Shoreditch one) where you won't get a Chicken Tikka Masala but you will get really delicious Indian food.
Lots of really good gastro-pubs all over London that live up to the name. The Eagle on Farringdon Road
http://www.theeaglefarringdon.co.uk/ was the original and is still good. Had a meal there a month or so back.
Definitely go to Borough Market.
5. Loads and loads of great local beers. Best Value are the Sam Smiths pubs. Princess Louise
http://princesslouisepub.co.uk/ near Holborn has an amazing interior and individual booths. All the drinks are from their own brewery....
Absolutely loads to do and see. Tick a few tourist boxes but get out and about and enjoy yourself.