How should I maximize soundproofing when installing new hardwood flooring?
There is no need to glue down the underlayment padding. The weight of the floor will hold it in place.  If you don't want it to shift while you are installing the floor, you can use double sided tapes (like the one used for carpet) to hold it.

I usually only put few strips of tape in "tricky" areas such as when approaching a door.  In a rectangular room, I don't use tape.