Flooring is typically installed first and cabinets installed over the finished flooring. But in an older home, sometimes there are compromises.

In my own house, about 13 years BEFORE we replaced the cabinets, we took up the old vinyl tile flooring in order to put in hardwood floors. Since this house was built in 1948, it had plank-style sub flooring throughout. The previous owner had put down plywood on which he could install the peel and stick tile that was there when we bought the house. So in order to make our new hardwood flooring level with the floors in the hall and DR, that plywood had to come up, but the cabinets were staying.

The hardwood flooring company had a devil of a time cutting out that plywood in order to install the hardwood floor (damaged the kick plates substantially but we just painted them black so the damage didn't show). Now, the cabinets were coming out and new ones were going in! My new ones are installed on the old plywood-over-plank flooring, and they were able to adjust the kick plates so that the somewhat jagged edges of the new hardwood were hidden by them. We just put the refrigerator on the plywood as I did not want to have to have the hardwood floor people come out and try to patch it in (location of refrigerator was moved) and refinish the entire floor. It works just fine.

When dealing with an old house that has been remodeled more than one time, one must get "creative"!

What you say they are telling you makes no sense! Cabs are put on plywood all the time; BUT according to my cab maker, there are types of plywood that work best as a subfloor. A floating floor cannot be put under cabinets, true. A floating floor will butt up to cabs and walls; then toe kicks and floor trim is put on top of the floor to cover the edges. That is done all the time. Try to do it that way to avoid quarter round in addition to trim and toe kick.

You do lose the height of the flooring from the height of counters/cabs with a floating floor, but even that can be fixed by putting plywood under the cabs to build up to the thickness of the flooring.

You need no flooring under the cabinets. Cabinet installers should be able to shim up the boxes so they are level and at the height that would yield the correct counter height once the floor thickness is taken into account.

Are you on wood or concrete subfloor? Perhaps the opposition to plywood is because you have a concrete slab and the wood would be in contact with concrete.

Yes, shim the cabinets to match or even slightly higher than the floating floor. Have your kicks left off and then the floor can be pushed underneath where the kicks would go. Have the kicks cut to length so you can install if you don't want to call back the installers. Once the kicks are on, it will look more finished. I do this all the time. You may still need a small moulding on any side panels, island back panels, etc. Typically a floating floor will need around 1/4" gap around any walls, posts, etc. You should also leave any base moulding off your walls for the same reason. Once the floor is installed, you can install the kicks & base moulding.

In my area it's much more common for the cabinets to be installed on the subfloor with the flooring NOT going under the cabinets, no matter what the flooring is (but it does go in the dishwasher, fridge, and range holes).

Are your installers also the floor installers?

Use IKEA and hang all cabinets (uppers and lowers) on a rail.

Thank you all. The current floor is wood joists placed diagonally. I think we are going to put down plywood, then the cabinets. Would it be ok to place the cabinets on top of the plywood and then have floor installers install floating floor to the cabinet and on the area wherwhere the dishwasher and fridge will be? Why should I level the cabinets with the floating floor?

"Why should I level the cabinets with the floating floor? "

It will give you a cleaner look if you can have the floor push underneath kick, panels, etc. But, you don't want them to rest on the floor.
More importantly, if you place the cabinets directly on the plywood & have the floating floor installed without shimming up the thickness of the floor, your finished height of 36" to countertop will not be achieved & the opening for such things as DW will be shorter.