I haven't tried STT's reformulated (newer?) cleaning products, but I think they had introduced some that were supposed to be better for this (I love their STT sealer, but wasn't thrilled with the original concentrated cleaning products). I use 7th Generation Disinfecting Bathroom Cleaner (spray) on the tub and shower surrounds in our three bathrooms. We use bar soap, and don't have any problems with soap scum. We do wipe down the walls (tile) with a microfiber cloth after showering (instead of a squeegee--couldn't find one I liked, nor was there a good spot to keep it in/on our bathtubs), which cuts down on build-up anyway. For weekly/bi-weekly cleaning-cleaning, I spray the 7th Gen on the tub surrounds, let sit while I wipe down the mirrors, vanity, whatever, and then wipe it off with a microfiber (I usually spray a little on the cloth as well). You can let it dry without wiping, but I didn't like the spots that were left behind on our large format, glossy white wall tile. Also, I'm not crazy about the scent left behind when the 7th Gen is left to dry (especially when it builds-up over time). I still prefer using Lysol Aerosol Bathroom spray for the tubs themselves, the shower pan in the guest bathroom, and the porcelain sinktops. I just find it to work that much better at removing any residual soap scum that might be sticking around (no pun intended! lol), or for obliterating hard water stains.

There really isn't anything labeled for "soap scum" cleaning that is truly marble safe. They are all acid based instead of pH neutral as stone requires. The best approach is preventative, as outlined above. The big one is to eliminate bar soap and many hair conditioners from your personal care routine. They will cause buildup on surfaces that can only be safely removed via old fashioned scrub brushes and manual effort.

I think these have come with use over the 20 years these countertops have been around. They are yellow discoloration marks in quite a few places. Then it is probably time to replace them. No way to know if those are burn marks, rust stains or chemicals from personal care items. I would try Bar Keeper's Friend or a stronger phosphoric acid cleaner (those used for removing hard water deposits.) If those don't do anything, then you can rule out rust.

I would try keeping it wet for about 24 hours....maybe put a sprinkler on it, then see if the vine pieces have softened enough to come off with a brush. If it was Boston ivy, those little "feet" have some sort of glue in them to allow then to climb even on glass, and of course rain doesn't wash it away. But perhaps the sticky bits are dry enough that they will absorb water and soften. You might try some sort of solvent....maybe mineral spirits....maybe denatured alcohol. Test on a small spot to be sure it doesn't also remove the paint shading.