I’d do something like a wall mounted network rack. Add some 2x4s to the shed wall for extra support and mount the rack to the 2x4s.
NavePoint 9U Wall Mount IT Open Frame 19 Inch Rack with Swing Out Hinged Gate Black https://www.amazon.com/dp/B071ZDXSGK/ref=cm_sw_r_cp_api_i_724EXQMV26W94QFAPVWM
As someone else said, fiber is the best for building to building applications. And as you mentioned, placed in conduit for any future upgrades or additions. Other pluses of fiber is future proofing. Also you’d only have one cable coming into your shed as opposed to several which also translates to only fishing 1 cable.
You can move AP #1 up and to the right - at the kitchen/dining threshold. More than enough range to get to the great room and closer to the outdoor room.
The rack will put out some heat and the fans will make some noise. So a smaller closet isn't the best idea. If you have a basement, I'd put it there. If the closet is the only place, then it will work. The 3rd option - if you must have it in a living area - would be the 2nd floor laundry room.
You can get inexpensive wall mount racks. Fold-out ones are nice. Leave a ~2' courtesy loop in the CAT6, so there is extra cable to work with.
Also keystone patch panels are really nice to keep the rack organized.
Monoprice has good prices on the bits.
Think about putting in ENT conduit (aka "smurf-tube") for the APs and the more difficult wall RJ45 locations. This makes pulling new wires possible. If not, run two wires to the APs for redundancy.
Thank you u/AncientGeek00! This was helpful. I went ahead and ordered a new rack from Amazon.
Fan noise isn't an issue, as my existing rack is in the basement.
​
I am using this rack:
https://www.amazon.com/NavePoint-Mount-Frame-Swing-Hinged/dp/B071ZDXSGK/
I know it’s probably cheaper to get them through Ali or Mono but I like to get my things a little bit quicker than they seem to get to me from those two. I’ve been paying the Amazon tax or I go to Micro Center.
NavePoint 9U Wall Mount IT Open Frame 19 Inch Rack with Swing Out Hinged Gate Black https://www.amazon.com/dp/B071ZDXSGK/ref=cm_sw_r_cp_api_glt_i_041X5FE3SJ750YN0B0RC
StarTech.com 8U 19" Wall Mount Network Rack - 12" Deep 2 Post Open Frame Server Room Rack, Black & 1U Vented Server Rack Cabinet Shelf - 10in Deep Fixed Cantilever Tray - Rackmount Shelf https://www.amazon.com/dp/B09338VB67/ref=cm_sw_r_cp_api_glt_i_V9QW0A2E0E9X7XQ09VVM?psc=1