We just had BR-111 brazillian cherry 5/16 installed on a slab. This was the first time this flooring company had installed 5/16 solid wood. The 2nd day after install floors started cupping just as pictured above. Some areas were worse then others. They sent out one of the flooring inspectors and he measured for moisture. Everything seemed fine and he could not figure out any moisture issues. The next day the owner came out. He brought out commercial dehummidifiers. He kept insisting it was moisture although no proof. My husband issisted on a qualified wood flooring inspector. He was to come the next day. The owner of the flooring place left. Less than 2 hours later he called on the phone and said he was sending some guys out to pull up all 1000 square feet of flooring in my whole downstairs. What was the problem??......... They had used the wrong glue. BR-111 rep said they had a new specific glue for this flooring on a slab. Flooring company had just assumed they could use glue that they usually used on all hardwood. The areas that were the worst were the areas that they had done the second day(thay were here 2 days laying floor) on the 2nd day they used a glue that was more like honey consistincy. These areas were really bad the flooring that was layed the first day did not look near as bad. If they had not changed glue the second day We may not have noticed. The other glue would not have shown the problem so drastically right away. it may have taken several weeks for the other areas to start cupping. This is the information that the certified inspector told the owner of the store. Needless to say he had 2 office personel out at my house tearing up 7500 worth of wood floor in there dress shirt, kakis, and dress shoes. They re-installed a couple of days ago with the glue from BR-111 and so far no problems. What a hassel it has been and I am installing a new kitchen.

Using a diamond blade, one could cut a diagonal line across the tile, either at the end of that handrail or somewhere between the handrail end and the start of the curve and remove a small portion of tile. This would tie the big room upper left to the small area lower right in your picture and you could use standard transition moldings. As to your tile grout issue, you might look into whether you could hire someone to clean and seal the grout or remove it and regrout it. Everything has it's drawbacks including wood. While it doesn't have grout and isn't prone to fracture cracks, it's more sensitive to water, humidity changes, dings and scratches. It should be recoated periodically. High wear areas will need recoating before low traffic areas. On the plus side, many wood floors can be refinished and you can change the color, unlike that pink tile someone thought looked great 15 years ago.

I found this on https://www.gsa.gov/portal/content/113602 Removing Linoleum Adhesive From FloorsProcedure code:965001SSource:Ohj - Shirley Maxwell & James C. Massey, 9/30/92Division:FinishesSection:Resilient FlooringLast Modified:08/02/2016 REMOVING LINOLEUM ADHESIVE FROM FLOORS PART 1---GENERAL 1.01 SUMMARY This procedure includes guidance on removing the mastic adhesive left behind when linoleum flooring has been removed. Linoleum flooring was very popular from 1863 to 1974. It was used historically as an original floor treatment in some GSA buildings. Linoleum's chief ingredient was linseed oil, pressed from flax seed, which became a tough, elastic material when exposed to air. The process essentially consisted of mixing linseed oil and gum with ground cork or wood flour, and pressing it onto burlap or canvas. It was then glued down to concrete or wood floors using a mastic cement. See "General Project Guidelines" for general project guidelines to be used along with this procedureThese guidelines cover the following sections: Historic Structures Precautions Submittals Quality Assurance Delivery, Storage and Handling Project/Site Conditions Sequencing and Scheduling General Protection (Surface and Surrounding) These guidelines should be reviewed prior to performing this procedure and should be followed along with recommendations from the Regional Historic Preservation Officer (RHPO). PART 2---PRODUCTS 2.01 MATERIALS Clean, potable water Vinegar Citrus degreasing solvent such as "Limo Sol", or approved equal. Dry ice Chemical solvent such as isopropyl alcohol, paint thinner, xylol, lacquer thinner, or paint stripper. 2.02 EQUIPMENT Heat lamp or hot-air gun Putty knife or scraper PART 3---EXECUTION 3.01 ERECTION, INSTALLATION, APPLICATION NOTE: A waterproof glue is usually applied at the edges and seams of the linoleum; the center of the felt layers is usually attached with a water soluble paste. Soak the mastic area in hot water for 20-60 minutes or until soft.\ If this is not effective, try adding vinegar or high-strength citrus degreasing solvent to the water and then soak for 20-60 minutes until soft. If mastic still will not soften, try applying heat to the surface using a lamp or hot-air gun. -OR- Try freezing the mastic with dry ice to break the bond with the substrate. If no other methods are effective in softening the mastic, test a chemical solvent on the area. NOTE: Be sure to test in an inconspicuous area to determine appropriate chemicals and strengths before proceeding with cleaning. Scrape away the softened mastic using a putty knife or garden edger. END OF SECTION interesting................

I say sometimes you have to cut your losses. Why not go to a habitat store and get you some cheap ceramic tiles and put them down. The ceramic tiles will withstand the cold and will also be better for a basement floor in case there is any moisture problems. It will be MUCH cheaper to go this route. Home Depot has deals for 20 cents a tile and in habitat you can get the 18x18 for the same price. I would use the big tiles because it is easier to put down and last sooooo much longer. In any case I think your tiles popping up is based on the wood subfloor. Wood floors constantly expand and contract. For your subfloor I would put down cement backer board. I used the thin one for my vinyl tiles. no shrinkage and moisture problems and my floor is smooth as smooth can be. Quarter-inch backer board is used for tiling over an existing solid sub-floor. There are all sorts of different tile backer board products out there. Some are cement-based, some are gypsum-based and others have a mix of ingredients. The cement-backer-board products can be a little tough to work with during the installation process, but in my opinion, they pay you back for the effort. To make your cuts score the cement backer board with a box cutter. You make numerous passes along the cut line, and then apply pressure to the other side of the cement backer board along the scored line. It usually snaps the backer board in two quite nicely along the line. Also I left a piece of backer board outside for the last 5 months and it has been rained on, in the sun, and in the snow. There is to date NO visible damage to the board. It looks brand new. That tells me it is very versatile and will withstand moisture and temperature extremes. If you insist on using the vinyl tiles again I would put down the backer boards for the subfloor. I followed some advice from others on using the glue on the tiles. It worked. It made the installation much easier and the tiles have not come up. My house is 70+ years old and I had a wood floor. I could not use ceramic tiles because of the weight of the tiles. There is no insulation under the floor so it is subject to the outside temps on the bottom and inside temps on top. No problems so far. Good luck! Hopefully you can turn this frustration around into a smile!! Here is a link that might be useful: cement board info