Use the Heathrow express not the tube to get to Heathrow. it costs more but is far faster
This, definitely. And as it goes from Paddington you can go to the Victoria, which is a 5 minute walk down London Street, straight across from Paddington. One of my favourite London pubs, it does great Fuller's ESB.
And if you like a good curry, the Noorjawan opposite is truly excellent.