Around here, our caliche is hardly affected by vinegar or muriatic acid. I'm not sure what it is, probably silica-heavy something, but my windows and toilet bowls sure could use an easy way to remove those deposits.
Same problem I have. Caliche is a nasty creature. I swear it has it's own life form to it. I can get a hole dug through it, and break through that layer, plant something, water well and I swear it reforms on the surface of the watering well of the tree. I periodically need to go in and break it up. Mind you my watering wells on trees are good sized and deep. Often six inches deep by a good five feet wide - and that is for a 5 gallon tree. Some of my neighbors think I'm nuts. But I plant mostly natives, water them really darned good, then wait several weeks before watering again so the roots go down towards water. My neighbors all have the drip irrigation they run on their trees for an hour every week and seem to think that is going to sufficiently water their trees and allow the roots to grow properly. I wonder how many of the Ash trees my neighbor has planted will go over in storms in the next decade because their roots stayed in this tiny itty-bitty hole?
I get the toilet rings all the time. I found a stone called a "Pumie" at the store marketed for taking scale off toilet bowls. It's pumice on a stick! It works too, no chemicals required. Vinegar and citric acid really haven't touched it. Despite using a squeegee on our glass shower doors, we still have buildup that I cannot get off no matter what I use - vinegar, citric acid, bleach, Citrus cleaner - none of it works. I'm starting to think it's something other than calcium myself. I'm tempted to make a solution of CLR to spray it with and see if that works. The problem I have is I am on a Septic so I need to be careful of what I use.
As for the rock - if you know it is solid jasper/agate/quartz, using the vinegar is a good option. I also second the motion of soaking it in baking soda and water to neutralize any acid residue that remain, then rinse off the rock afterwards. Goodluck and show us what you find under that white stuff!