Yes you can tile or lay engineered hardwood over concrete.

They are a great base for any future floor. If you want concrete it will have to be ground down and restained.

It should only be leveler. I would just hammer it out. Trying to refinish would be a lot of work. And may still not be successful. Stain will penetrate as much as 1/2" deep.

I would suspect it would be similar to Terrazzo where you can glue an engineered hardwood over it providing waxes are removed and the Terrazzo scarified for a good adhesive bond.

I would think you could lay any glue down flooring directly on the concrete. Is that a wood floor in the dining oom? it probably has the same concrete base.

The epoxy and dye stain are bond breakers for adhesives and thin-sets. It needs to be removed. You can always float over it.

Grinding is pretty easy with the right equipment- your contractor can even rent one (dont use Home depots though unless they give you new diamond pads)- there are allot of decorative concrete companies out there - you can also try garage floor companies as they have grinders. It may be hard to get a stain down again as it may come up blotchy so a metallic epoxy or alternate may be in the cards.

That DR looks as though they did exactly that a floating floor over the concrete.

As others have stated, it is pretty tough to get another floor to 'grab' a finished concrete floor like this. To get tiles to go down properly, you need to shot blast this surface, prime, then apply new concrete. Once that's done ($3-$5/sf) and all nicely leveled up, you can then look into adding tile. That's when the regular price of $12-$15/sf comes in.

A floating floor is possible. A vapour barrier can be put down over the existing finish and then you float whatever you want over it...assuming the concrete flatness rating is good enough. If it isn't, you will need to get the flatness to a proper level (see the 'grinding/blasting' option).

A stained concrete floor is notoriously difficult to get rid of once it's there.