Take some tips from the flooring manufacturer as to what is needed to provide for an acceptable substrate for their tile. The product ought to come with installation instructions or their website will give you what you need. Since this it is a click-type floating product, you don't need to be concerned about getting your plywood seams tight or sealing them. Just fill in the missing pieces and fasten it all down. You can't get fasteners in under the cabinet toe kick without the right tool, but that is not a big deal. Just fasten as close as you can to the toe kick; the weight of the cabinet will hold much of it down anyway. If the adhesive residue from the linoleum is "sticky"...then laying down a single layer of newspaper or craft paper will avoid snapping noises as you walk across the floor.

Are you my doppleganger, Mable? This is almost our exact circumstance, except we live here and don't rent. We are in a small ranch (in Alaska) and tore down the wall between the kitchen and the living/front door area. We are also on a slab, which limits us to engineered wood if we go with real wood. Almost universally, the local flooring folks are recommending LVT or laminate over the engineered hardwood - even though it would give them 3x more of our money if we went with real wood. They all say that the laminates will wear better/scratch less than the wood for the same water risks and are better with animals or men who don't take boots off at the door. Some of the newer laminates have much better systems for water-resistence, too. We are seriously looking at LVT's, which is a market that has exploded a lot over the past year or two. There are now loose lay LVT options, which are great because if a plank gets damaged you can just literally use a plunger to pull up the damaged board and replace it. There are also floating, click lock systems for LVT now. (We really don't want to glue down after pulling up all this crap from our foundation and hearing horror stories about glues interacting with concrete subfloors, even though glue down has distinct advantages in terms of not needing such a level subfloor and sounding less hollow.) We are still on the fence about it all, but my partner wants to go with a loose lay LVT floor (Kardean, XL Flooring Drop and Done or EZ-Lay). I don't love the look of the LVTs and am leaning towards laminate and taking the water risk to get a little bit of sheen and I think they reflect more light for our small home. I think the laminates look more realistic to wood, too. Currently, I'm researching a product from US Floors called CoreTec One and CoreTec Plus, which is marketed as a laminate with 100% waterproofing that is able to go over uneven surfaces.

This forum is best for things like landlord or neighbor disputes. You might want to try more relevant forums like Flooring or Remodeling or Home Decorating.

Have you looked into COREtec? Menards sells it. http://www.usfloorsllc.com/product-category/coretec-plus/tile/