advice re acoustic treatment in new home studio

HI everyone,

this is my first foray into gearspace and after reading through other threads this has already been very helpful so thanks you in advance!

Im a pro sax player and currently converting my garage into a home studio for teaching, rehearsing and home recording and music production. The main activity in the studio will be for practicing and teaching saxophone with occasional full band rehearsals and then finally and to a lesser extent recording and production. We are in the final stages of soundproofing so I am now in need some advice re acoustic treatment for the room.

The room is 4 x 3.6 meters so not quite an even square. The ceiling height will be roughly 2.3 meters. Due to my budget restrictions I'll most likely be doing the acoustic treatment of the room in stages.

For the walls I've purchased 6 rectangular panels [125cmx65cm] that Ive bought off my buddy who is leaving his studio space. They're fit for purpose and built with pine frames /rock-wool interior and covered in cloth. I know I'll need a few more for the walls but I am starting with these. One of the walls has a window where Im hanging a heavy/acoustic floor to ceiling drapes that will nearly cover the entire wall.

What I really need is advice about the ceiling and corners.

For the ceiling my man in charge of the build has offered to design /build and install 2 large acoustic ceiling panels made with similar spec to the panels Im putting up on the walls. As We have down lights being installed we are discussing putting the down lights though the acoustic panels so as not to obscure the lighting or cast weird shadows in the room. My main question is this.... I know there needs to be an air gap between the ceiling panel and ceiling but can you affix the wood frame directly to the ceiling and leave the air gap between the rock-wool inside and the ceiling or do the ceiling panels need to be hung with hooks so the wood frames are not connected or touching the ceiling at all? The builder is saying it will be much easier to install and look nicer if he can affix the frames directly onto the ceiling but leave gap on the inside. Its worth considering Im not doing almost no mixing and mastering in this space. The main activity in this room will be sax playing with occasional full band rehearsals and then recording for sax for producers songs etc. My uneducated hunch is given the small size of the room and relatively low height of the ceiling that the wood frames could be attached to the ceilings even though I'm aware this is not the very best for acoustic design.

For the corners … Im aware I need some treatment so looking into bass traps but given the main activity of the room Im feeling like this can come last when I have more money. I have foam bass traps I bought on eBay years ago that Ive treated another room with where I teach online and practice from home. I know the density of this material is not quite right for proper bass traps but Im thinking I could put these in the corners to start with and when I have more money either build or buy some bass traps fit for purpose.

Just a few questions I had is do the bass traps in the corner need to go floor to celling? The builder is suggesting if we have a lowered or hanging ceiling that it wont look as nice aesthetically if the bass traps in the corners go all the way up to the ceiling. I was recording in a professional studio recently where the bass traps went from the floor but didn’t go all the way up to the corner of ceiling. So I did wonder about this. Also given the smaller size of the room I don’t want to install really big traps in each corner as it will make the room feel even smaller.

Any advice on any of this would be hugely appreciated! Thank you!