Heathrow Express is much faster than Heathrow Connect but costs 10 GBP more. Plus, with the Express, you have to change trains at the first Heathrow stop to get to Terminal 4. Heathrow Connect goes direct.
Yes, I meant a taxi to Paddington, not Heathrow.
I've stayed at the Vanderbilt back in 1993. It's right on Cromwell Rd ( a major thoroughfare) so if noise is a problem, try to get a room in the back (but they did have big metal shutters that helped to keep the noise out.) It's across the street from the Gloucster Rd. tube station. Great area for pubs, food shopping, little restaurants, etc. (My favorite street.)
I've heard great things about Base2Stay and plan to try it later this year when I'm in London. It's only a few blocks from EArl's Court station.
The Sumner is not in that area. It's near Marble Arch--near the NE corner of Hyde Park.
The Ashburn is a couple of block from the Vanderbilt. I've never stayed there but have passed it many times.
I'm not familiar with The George but it's a few block from Earls Court station as well.
Traveling back to this area after the theater is no problem. I do it all the time.
As to what time to leave, check with the hotel and the taxi company. They would know best.