Clemburt
You are correct in your understanding, to do Ian's test ensure your AEBS is connected to the BT HH when you power up all your Sooloos equipment, the reason I say this is that when I had the same setup as you my BT HH was the DHCP server (the one who gives out IP addresses) once everything is working disconnect the AEBS from the BT HH or disconnect the switch from the AEBS whichever is easier? As for your new desktop pc, you can connect wirelessly to the AEBS or wired to the AEBS, try to avoid going via the switch obviously!
Cheers
Russ