The most effective under-deck insulation
I have come across is comprised of multiple layers of thin bubble-pack with aluminized Mylar skins. Many adhesives would work to hold it to the overhead, including contact cement. I would completely fill the available space, if possible, leaving no air gaps.
Lots of choices on headliners. Consider thin, laminate covered plywood panels
with as few battens as possible covering the joints and edging; use minimum number of fasteners; leave screw heads showing. Durable, easy maintenance
, easy access. Similar installation
could be done with only heavy gauge laminate, no plywood
, if the spans were small enough.
Padded headliners cut down the noise