If the rings are dried scum from the vegetable and brine, use elbow grease and hot soapy water (with ammonia added) to remove. If the rings are actually etch marks in the ceramic finish, nothing will help.

One makes pickles by following a recipe and unless specifically stated otherwise, the vinegar used is "standard" or "5%". Read the label carefully and do not use 4% vinegar for pickling. The recipe will give the amount of vinegar to be used in conjunction with the stated amount of brine, as well as how much salt is used to form the brine (lots of variance depending on the vegetable and the seasoning).

Back about mid-century my grandmother had pickle crocks and she insisted that they should never be washed nor used for anything else. Having said that I think she may have rinsed them with plain water and a rag. She had separate crocks for dill and some other kind of pickle she made.