You can get them at any of the underground stations and some shops as well I believe. There's no reason to not get one as it will make your transport easier and cheaper and you can get a full refund of your deposit and any remaining balance when you leave. You can use your card pretty much everywhere, however some smaller shops may have minimum spends (typically £5 or £10) so having a bit of cash on you is good if you want to get a bottle of water or something. Tube is usually quicker, but taking the bus, especially as a visitor, can be nice as you see the ground that you pass.
6 days gives you plenty of time to see some of the 'other' things to see. Some things I would consider doing:
Take a public transport boat down to greenwich and then go to the meantime brewery for a tour. Eat dinner in bermondsey at either Zucca or Jose (book either far in advance).
If you catch some nice weather, starting from the west side, walk through little venice and then have a dim sum lunch at pearl liang (look up how to find it before you go as it can be confusing) or down some southern US food at the lockhart.
I would not miss St Paul's or Westminster Abbey. You might need to go very early to Westminster Abbey or risk waiting hours in line. I would allow a few hours or more for each and you should go up to the dome in St Paul's. I think the stairs close at 16:30 so plan accordingly. I'd consider a burger for lunch or dinner at Goodman City or a full meal at St. John when you do go to St Paul's.
The British Museum is also a must. An (long?) afternoon is probably sufficient, but if you're really into that kind of thing I imagine you could spend days in there.
I'd see if you can score some tickets to a performance at Shakespeare's Globe. They're probably sold out, but you might get lucky and find some returns or standing tickets.
The National Gallery is also really good. Have a coffee at Notes on the nearby St Martin's Lane. For lunch, if you go on a Thursday there is a market in Covent Garden square, otherwise consider: Kopapa, Da Polpo, Green man and french horn, or flesh and buns.
Take a day trip out to Hampton court palace. Not sure where to eat here, but I'd recommend avoiding their cafe. I think you can picnic in the garden though.
I'm not a huge fan fan of Camden market. I think it's overcrowded, touristy, and full of junk. The high street isn't much better. I'd say go to one of the east markets instead. IE
http://www.broadwaymarket.co.uk/ Another option is to go to Borough market
http://boroughmarket.org.uk/ . Both zucca and jose are nearby, but you can also go to wright brothers for fresh seafood and oysters (get the pint of prawns). Have a coffee at monmouth.
The London Eye isn't really worth it, though it's not unpleasant. For the money, I'd say you're better off going to the Shard, the heron, or one of the other towers with high floor bars and getting a couple of drinks instead. You could also eat at one of the restaurants in the same towers. All definitely include a premium for the view though and I would avoid sushi samba. My opinion of Duck and waffle is up in the air, but most people seem to like it (food is reasonably priced, but drinks are extortionate). I have no experience with the restaurants in the shard, but everything I read seems quite positive apart from the prices.
If you're into parks and outdoor spaces you could go to Hampstead Heath (though I would definitely visit Hyde Park and St James Park). Hampstead is quite a posh area and both the park and surrounding streets are nice to walk around. You could also visit Keats house. You do get a view of the city skyline from the park. I would recommend getting lunch at Paradise Hampstead(indian - amazing and cheap). Oddono's has good gelato as well, but Gelupo has a leg up on it.
As I mentioned before, get gelato at Gelupo and consider street feast as well.
Just as a side note, be aware that picking a random restaurant in places like Leicester square can be a bit 'risky' as a lot of them are tourist traps charging just as much for mediocre food as other places charge for good food. That said, most of the chain lunch places - Pret A Manger, Eat, Itsu, Abokado, etc are all pretty good and a great option for a quick lunch.
I don't know if I'd rate these suggestions in any particular order, and they're by no means exclusive, but I hope they give you some ideas outside of the 'main' stuff.