HEX comes in at street level. IIRC, there are some escalators down to the tube, but I can't remember if they go all the way - its a bit academic, at St James', it's steps only I'm afraid. I would be tempted to go for a taxi, but they can be in short supply at Paddington, particularly first thing. OTOH, the queue is well regulated. I'd guess it would be less than £10 for a taxi from Paddington, its a shorter journey than from Kings Cross which is about £10-£14. Of course, in London, these things are always traffic dependent!

If Shepherd Market is close to the Ritz, that puts it in the Piccadilly area. map of the area, with the postcode highlighted is here - the street name itself is not too clear. There are some interesting little restaurants in this area. To get there, you could either come out of your hotel, and walk across St James Park and then Green Park, or come out of the hotel, walk to Victoria (you could get on the tube, but you'd only need to change at Victoria from Circle & District onto Victoria, and trust me, its quicker to walk from that hotel!). Quickest way to Victoria is to come out hotel, turn left onto Buckingham Gate, first left into a little street called something like the Groves. At the end of that street, either turn left again and then right at the traffic lights then walk along Victoria until you reach the station, or if the building ahead of you is finished yet, there will be a pedestrian walkway through to Victoria - turn right when you reach the street of Victoria. Either way, its less than 10 mins to Victoria (station). Once there, get the tube to Green Park (probably Victoria line).

Not sure about the 7 day passes, I've switched onto an oyster card myself.