Looking at your setup I would look to do the following setup all in Ubiquiti:
House - USW-Aggregate for 10GB link to shed. Any devices you want connected at 10GB can be done via DAC cable or Copper Modules.
House - USW-24-PoE - Passively cooled POE switch (16 Ports) for Access Points etc. Connect this at 2GB to Aggregate switch using 2 x DAC cable due to 1GB port limitation, you will need to manually configure port speeds on each end.
Shed - USW-Aggregate for 10GB link, mirroring the house side
Shed - USW-24-PoE - Similar setup to house
Adds a lot more ports than you require but future proofing and is based on their currently available hardware.