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).

Hard water in and of itself is it's own problem. Mildew in the grout is usually moisture being held in the wall. Today's construction methods; topical membranes, better ventilation, better grout formulations, should eliminate that. Streaky walls? I'm a bar soap guy, for years we used Zest which had no oils. We did absolutely nothing to clean our shower other than a brushing down maybe every 6 months or so. If even that. A few months ago we started getting residue on the walls. Checked, a sure enough Zest reformulated their bar soap. Same holds true for hair care products. Watch the labels. Dark tile will show more than light tile. Glass will show everything. If you go with a hand held shower on a hose, you can use the hand held to rinse down the walls. That'll prevent "dirty water" from drying on the wall and leaving a hazy film of product residue behind.

Depends on the scratches, if they're scuff marks you might be able to buff them out. Some small scratches you can take out by mixing baking soda and water but make the paste real thick and apply over the scratches and rinse them off. Some scratches are there to stay and if you have a few spare tiles and scratches are real bad, just replace a few damaged tiles. Good luck

Oh I’m sorry they put them in the shower. I only have them on my bathroom floors. I am currently using Bona products. They are Ph neutral and readily found in retail stores. When the cement tiles need more help than regular cleaning, you’d be surprised at how aggressive you can get. Both my floors had different installation issues and needed to be restored. In the first bath the installer got thin set on the tiles before they had their first coat of sealer. I had to sand them all down before we could grout. The second bath had sealer, but the navy pigment in the grout stained anyway. In the end Mr Clean Magic Erasers scrubbed them clean without damage (they are chemical-free and made out of melamine foam) You can even use peroxide safely. Check out the cement tile websites for more care tips.