Unfortunately, it does not fit the Lite.

No. Just buy a bluetooth adapter for your PC.

The manufacturer's website only shows the Switch, but it should also work with the Lite version, being plugged into USB-C and the console was basically the same.

We have already placed an order. Unfortunately, we currently have no information about when the item will be available again and when we will be able to deliver it. The supplier himself is still waiting for a subsequent delivery from the manufacturer and for this reason he had to set the delivery date to unknown. As soon as something changes in this regard, we will post it here. Sorry.