Hello nordy. The license will not be lost when you reinstall the app.
The license is bound to your google account and will be available on every device you are using with this account.

The auto detection of your devices only works if your network router/hotspot supports “multicast”.
If the direct connection via ip-address also doesn’t work, there might be an active firewall (on the router or one of your devices)?!
greets
jens