Baker Street East

re 5, Generally the pattern of railway station naming in Britain is that qualifiers before the name of a place signify that it is separate from the place, qualifiers afterwards are for distinguishing more than one railway station in a place.

Taking the fictional town of Putware, a station called "West Putware" would be west of the town. Possibly in a village called "West Putware" with the village name meaning "settlement west of Putware".
A station called "Putware West" would be a station in the town Putware that was west of some other station in Putware (perhaps "Putware East", its self distinct from "East Putware").
This leads to possibilities such as "West Putware West" or "West Putware East" (although you'd hope that the train company/companies would be more inventive than that and name them something like "West Putware High Street" and "West Putware Riverside" perhaps).

There are of course exceptions to this of course, but "Baker Street East" will be signifying that the station is east of "Baker Street" station, not (in an area) east of Baker Street (the road), which it might not be (although in this case it is)

2. The Bakerloo had a building on Baker Street, roughly opposite Melcombe St (the road to Marylebone). It was a standard ox-blood terracotta Leslie Green station with lifts down to the platforms. It connected to the Met line platforms at Baker Street East via the footbridge at the north end (still in use, but just to link platforms 1&2 with 3&4). I don't think that Baker Street East ever had a separate entrance. However, the Circle/H&C platforms had their original entrance at the junction with Baker Street, where the subway now crosses under Marylebone Road, at the west end of the station.
3. I'm not sure whether Baker Street East was ever an official name - more of a designation for some platforms that were slightly further away. As such, there wasn't ever a renaming.

To answer your last e-mail, there are three gatelines at Baker Street. One is in the Bakerloo/Jubilee ticket hall (west end of the station, leading to the escalators). Another is in the Met ticket hall (central part of the station, under Chiltern Court), and the third is south of the Marylebone Road in the subway entrance to the station. This latter leads into the footbridge over the east end of the Circle/H&C platforms. I don't ever recall an extra set of barriers (King's Cross certainly had these though).
The Circle and H&C platforms were the original ones on the Metropolitan Railway, and date to 1863. You are right in stating that the current Met line platforms are on the site of the original Metropolitan & St John's Wood Railway platforms (although the layout has changed a lot over the years).