1. If you're going to be here for less than 6 months, I'd say no. I was in the UK for 4.5 months in 2005 for study abroad, and never had a bank account and managed just fine.
2. Bank of America!! They have an agreement with Barclays, so if you use Barclays ATM's in the UK you don't get charged fees. I have a BoA account and this has been a lifesaver. Highly recommended (even if you just get a basic checking account for this trip, it's worth it!)
3. You might have some issues, yes. Merchants are *supposed* to take cards w/o a chip&pin and swipe them, but in actuality many shops have just gotten rid of their swiping systems altogether and just have a chip&pin, or aren't familiar with the rules and will just refuse to take your card. That's not the case everywhere, though- many big retailers (think Marks & Spencer, etc) and some of the smaller ones will happily swipe an American credit/debit card. I tend to avoid using my cards here, though because of the currency conversion fees- if I need to use money from my American account I just take out cash. I lived almost exclusively on cash/ATM withdrawals throughout my semester study abroad, and it worked out pretty well.
4. Does your $6,000 figure include your accommodation fees? If it doesn't and you have $6K in spending money, I'd say that's definitely enough. If it does include rent money, then it's on the low side but still doable I think if you live on a tight budget.
edited to add: that's from my experience outside of London, though. London (as I'm sure you know) is miles more expensive than anywhere else in the UK, so keep that in mind...hopefully a Londoner will be able to give some more realistic figures.