I had a situation some years ago with some really, really dirty vinyl flooring. After trying a bunch of household, all purpose type cleaners and a lot of elbow grease, I bought some commercial 3M floor stripper. The difference was absolutely amazing; I put down the stripped waiting a few minutes, and years accumulated grime wiped up effortlessly. Unfortunately the stuff isn't cheap and you'll need to find a janitorial or industrial supply house. Well worth the money and effort IMHO. Here is a link that might be useful: 3M Floor Stripper

Your installer installed that with too small a space between the tiles, and didn’t clean the mortar out during setting You’re going to have major issues here. There’s no room for a standard grout. You’ll need to hope that Flexcolor CQ will work. But not until that mortar is cleaned. Which may chip the tiles. Because the spacing is too small.

This easily cleans grout. I had serious grout issues in my kitchen and utility room thanks to two huge dogs and I tried every solution possible. One day I thought I'd try the Clorox spray and I couldn't believe how easy it was. Spray the grout, let sit a minute or two, then scrub with a toothbrush type of brush. Then rinse clean. I also have painted cabinets and Dawn in warm water works good but not on the cabinets above the stove. Using a microfiber cloth to wash the cabs and another microfiber cloth to dry works miracles.

Mix a srong solution of Oxi clean and water. Wet grout lines and leave on for 30 miutes or longer. Keep wet with Oxi mixture. Then scrub with a brush and rinse well. I use a grout brush attahed to a long handle, saves the back. Purchased it from a cleaning supplier.