What is the white deposit on the inside bottom of glass vases, toothbrush glasses, my dish drainer,
Those are minerals (usually calcium of some type) that were dissolved in your water. As the water evaporates, the calcium can’t stay dissolved in the water, so it comes out of solution and leaves a deposit.
and is there something simple that would dissolve it?
As already mentioned, vinegar is one solution. Stronger acids are another, but can be nasty to deal with.
I have stumbled on a different fix. I leave a pitcher of water on my counter to top off evaporation from my fish tank. I’d let it sit for up to a week to let the trace amounts of chlorine the water company puts in the water come out of the water. (Chlorine is bad for fish) Since I have to leave the water open to the air, there would always be a bit of evaporation. And that left lots of hard water deposits on the plastic pitcher. A while ago, I started running that make up water through a Brita water filter before letting it sit out. In pretty short order, all of those hard water deposits disappeared. I figure that the water filter was removing a lot of the hardness (the dissolved minerals) from the water. That much softer water was now dissolving those old hard water stains and removing them for me with no work on my part. In a couple of months, the pitcher was as clear as the day we bought it.
You could replace the filtered water with distilled water. It should have the same effect. Try filling the vase or glass with distilled water, then wait a week or so. See how much of the stains disappear. Dump out the water (use it for plants or something similar), then repeat another time or two. My guess is that the stains will get rid of themselves with little or no work on your part.
I can remove the line in the toilets by scrubbing with a pumice stone. My daughter tells me that doing this will scratch up the bowl and make the problem worse. Is this true?
No. Pumice stones are safe to use on porcelain. You don’t want to grind them in with heavy pressure. Just press a bit more than lightly on the stone and rub it back and forth, keeping the stone wet. There will be no damage to your porcelain bowl. Once cleaned of the ring, keep the ring from coming back by cleaning the bowl every 2 - 4 weeks with a bowl cleaner.
I used a pumice stone on our toilet bowl when we first bought the house almost 25 years ago. It’s looked great since then - even through the times we weren’t the most diligent house keepers.