The UK government provides all the information you ask for (and more) on gov.uk. To surmarize what you need to do, from that site:

If you are entering the UK from the EU or a listed non-EU country your pet must:

• be identified with a microchip
• have received a rabies vaccination followed by a 21 day wait
• be accompanied by the relevant documentation
• be treated against tapeworm (dogs only)
• enter with an approved transport company on an authorised route

The US is one of the listed non-EU countries. The descriptions of the documentation are later in the linked page. The authorirzed routes can be found here - broadly speaking, fly to Heathrow, Gatwick, Manchester or Edinburgh, although that will probably make up most of your options from the US anyway.

I'd advise against trying the "fly to France then drive to the UK" option, because:

• You'll need to deal with the paperwork and beaurcacracy that will still exist there, but now in a foreign language (it may be laxer, but it won't be non-existent)
• You'll need to travel to the UK while still following all the above rules (unless you smuggle your dog out of sight)
• Your dog risks being an illegal resident of the UK. It seems unlikley they'll ever catch up with you, but I imagine the consequences are unpleasant if they do.