Welcome audiophyllic,
When we were preparing to build a new home 3 or 4 years back I innocently asked about including a HT. The answers I got back were in the $50k ballpark and all looked like variations of small, but lavish theater. That blew my budget as HT is a low priority for me. But I did build a nice dedicated audio man cave for very cheap:
First is getting the room proportions right. Like most, I was limited by an 8 foot finished ceiling height, so I followed the Cardias ratios and ended up with a 13 ft x 21 ft room to minimize standing waves. (I also use the Cardias speaker/listening location recommendations.) Your left over space could be used for storage/infinite baffle volume, built-in cabinetry, bass traps, etc. Your current 10 x 15 x 31 dimensions should be "adjusted" IMO.
Second was physical isolation. Interior partitions were built as staggered stud walls with batt insulation. If you really want to crank bass you could even go up to insulated 2x6 double walls with the studs at a varied spacing of 8 - 16 inches. I used an insulated fiberglass exterior door and insulated flexible ductwork to the room. My mistake was the ceiling (recessed can light fixtures and the drywall attached directly to the floor structure). The builder got snotty and would have voided any warrantee on the entire drywall job on the house to "float" the audio room drywall ceiling as I'd specified (and is common for such rooms). An acoustical drop ceiling with batt insulation above would probably have worked nearly as well, but we were running out of money. My walls/door have worked extremely well (the family can really crank the HT just a few feet away (no door between) and I can't hear a thing with music playing). But I hear the washing machine, vacuum, and footsteps above. In any case, don't penetrate walls or ceiling.
Finally is electrical isolation. Use dedicated 12 gauge/20 amp circuits for the equipment (wired into to top of the panel if possible). I had these dedicated circuits run to their own common ground. My one "splurge" was to buy cryo'd hospital grade receptacles and to have one 20A circuit used per each of the three receptacles. (I'm a "Keep It Simple Stupid" kind of guy, so the audio system is purist, simple, and small.)
All these extras cost me about $200 (as the builder insulated interior walls anyway and the door was cheaper than the solid cherry doors in the rest of the house) plus the loss of extra storage space (that wifey would have spent money to eventually fill). The isolation is so complete that at first is was a bit spooky. Now I can listen 24/7, at lower spls, and we keep better domestic peace. Easily the best money spent on the house and a great example of the value of planning ahead. Other than the ceiling, the only other worthwhile upgrades would be double 5/8 inch drywall (especially for loud HT bass).
Since then I've added six 2x4 high density fiberglass absorption panels at front corners, front wall first reflection points, and side wall first reflection points. I run single driver speakers and will be playing with a Behringer DEQ2496 I recently picked up to replace the baffle step/zobel circuit to allow for a direct monoblock to driver connection and to tweak the room/speaker response.
From what I understand the ceiling height should be close to the height of the tall line array speakers. This may cut down further the size of the room.
