Ah, typically I prefer to have as many cable drops as I can going to my network cabinet. I have two active and one inactive drops per room, with more where I have TVs and APs. While using a wired network connection is very low latency, chaining switches does add a very small bit of latency as the tree "expands" its branches. Having everything going to one place also aids in re-wiring stuff as needed via patch cables. Ofc, as you mentioned, you only have one drop to each room, so this won't be applicable in your case.
You will be able to put the switch in front of the Lyra AiMesh node or vice versa. It should work either way since the Lyra's ports will act like a normal switch when in AiMesh mode. I had to do something similar for the AP covering my backyard, as I didn't have the foresight to run a drop there originally. I ended up using the closest AP's spare port with a PoE injector (the APs are PoE powered, but the second port is not) to get WiFi coverage via another AP in my backyard.
You will be able to put the switch in front of the Lyra AiMesh node or vice versa. It should work either way since the Lyra's ports will act like a normal switch when in AiMesh mode. I had to do something similar for the AP covering my backyard, as I didn't have the foresight to run a drop there originally. I ended up using the closest AP's spare port with a PoE injector (the APs are PoE powered, but the second port is not) to get WiFi coverage via another AP in my backyard.