Although not v. experienced on all this, I did my first cross-channel a few weekends ago to Le Touquet and had a great day. Taxis wait to take you from the Airfield into town (about £5), and most of the shops and restaurants are open on Sunday. I didn't go for Duty Free, but picked up some really good wine. I didn't see the big sort of Supermarkets that I believe people go for, but a smallish supermarket was open.
I had absolutely no customs checks in or out.