Nice to see your bike has found a good home. Now for the funny bit. All of your questions can truthfully answered yes, no or maybe. Each is correct. Some D1's had batteries, some didn't. Exhaust systems varied slightly over the years as did the chrome plating or painting of various bits and bobs.
To give a precise answer to everything you first need to identify exactly what Bantam you have. Saying you have a D1 plunger 52 is not enough. D1's came in slightly different varieties.
There are various websites devoted solely to Bantams that can give you an exact history of what was done when and what the relevant frame and engine numbers are. I would first Google them and see what they have to say.
Secondly I would then check to see if what you have is original. Bantam parts are broadly interchangeable and after nearly 60 years on the road it is a fair bet yours is a bit of a mechanical mystery bag. This is normal. You may have all correct Bantam bits but are they all correct for your version. There is no point fussing about paint or chrome if you don't have the right part anyway.
The Bantam is an ideal bike for a rolling restoration. Use any means to get it on the road first and then start working on getting it right bit by bit as you go along. People are always much happier to help if they see the bike on the road and being ridden.
Best of luck,
