Cashu Pay Server on Nostr: Yes, you can cash out automatically to a lightning address (not on-chain currently, ...

Yes, you can cash out automatically to a lightning address (not on-chain currently, although as mints support on-chain, this will be possible too!).

If you have woocommerce store with a btcpayserver plug-in, it works out of the box, yes, precisely because it is compatible.

For this specific scenario there is also a WordPress plug-in that will setup cashupay and configure the btcpayserver plugin automatically, so you don't even have to upload anything manually, just install the plugin and your integration is autoconfigured.

For other services you use the btcpay plugin compatibility exactly as you described.