To answer your questions in order:
There are several ways of searching. I have found the easiest way is to search by keyword. Then you can type in the surname you are researching. You can also narrow down the search by county or decade.
It is hard to say whether they had an actual shop or not. I would tend to think so, by advertising in these directories. One thing you must remember is that just because they are not in the directories, does not mean they weren't still in business. To put your name into the directories was an added cost.
I can't answer about the Birmingham library; but don't rule out local history and family history groups, or even the nearest archives. Any of them could hold photos.
Try a google search for "photos of old Birmingham"; I came up with a lot of hits. You might get lucky that way.
And no, it doesn't give the name of the building, judt the address.
And thanks Stan, I was still typing when you posted; I edited before I posted it.