Which travel sites are cheaper to book from for accommodation in Europe/England? Which credit cards tend to have further cashbacks and discounts (Agoda has relatively cheaper rates from my research)?

I think you can refer to the various booking sites to search and book directly with the accomodation. It will be cheaper this way.

If you're looking for really cheap accomodation, you could consider staying in a hostel? Hope it helps!

I always use Shopback and Expedia. Expedia rates might sometimes lose to booking.com but because I stick to Expedia, I have became their GOLD member which results in more Expedia points per stay. All these points can be used to pay for my next hotel booking so it's a huge win! I use SC Manhattan or Spree as they have ongoing promo code with Expedia too!