You want the fastners to pull down the underlayment for 100% contact with the the subflooring, there's always a tiny sag between the joist.
I'm hoping you miss understood them, there is no "cement adhesive" that can fill in the gaps or low spots.
Screws are a waste of time and money on this one, as the screw goes in it's going to lift up the underlayment unless it's predrilled and counter sunk, there going to crush the plywood and leave a raised spot.
No the staple gun will not fit under the toe kick, but there's just no need for it to be attached there because no one will ever be standing there.
Just by the things there suggesting will be OK I'd be very concerned if they have a clue what there doing.
I'll make a guess, you mention this to them and there going to say this is the way we have always done this.
Doing something wrong for 20 years does not make it right.
In the past I've been paid $1000.00's to go back and fix so called pros screw ups.
Adding a third layer of shingles over a leaking roof, building a deck even with the threshold and attaching it to the house with no flashing, another one used 1/2 tile board on the floor and used 1/4 on the walls.
I spent my first two years in business doing nothing but going behind just one contractor that had been kicked out of the county.