Sorry I didn't get back in on this. The grandchildren arrived for an overnight. Minor bedlam.
Yep. Any adhesive would work. Yes to the short screws also. The floor underlayment is layered to hold fasteners well. Cost? Here in S. Jersey a 4 x 8 1/2" birch is around $45, I pay $30 for 100 sq. ft. of the foam at a flooring distributor and the 4 x 8 underlayment is $15. So $75 for a 4 x 8 sandwich.
!/2" birch 4 x 8 $45, 4 x 8 Homosote $27 +$72 and it weighs a ton.
The weight of the homosote deterred me from using it. If I were to build a permanent layout, I really like the cork road bed only and just scenic the plywood everywhere else. A 100% homosote is not really needed. The track roadbed is what needs to be sound absorbing material.
I am building portable modules, so weight is a big concern. The temporary holiday layout that I am building will have the sound foam with white felt on top for snow. That will be my test to compare it to homosote and the ability to hold a normal volume conversation with the trains running. If the results are positive, I will use it for the modules.
My target for my modules is March\April.
The sound experts use the same closed-cell, cross-linked foam in greater thicknesses. 4mm in airplane passenger areas, custom cars, etc. I was intrigued by it's light weight and performance and looked for an everyday product.
As I said before, it all works, as others have attested from experience. Use what's available locally at a cost that works for you.