Is it possible to have a NodeMCU running on a different network?

I have a NodeMCU at my girlfriend’s place and I would like to connect it to my HA installation at my home. I setup an MQTT broker and connected both HA and the NodeMCU to that broker. Should the device get automatically recognized/added as a device by the MQTT add-on? Or am I doing something wrong? Is it possible to update the NodeMCU’s program over the internet (without my girlfriend having to manually flash the .bin everytime I update something)?

Does ESPHome support discovery for MQTT?

It’s a long time since I used MQTT and ESPHome but I vaguely remember having to manually create MQTT sensors in Home Assistant.

I hope you are using securely encrypted MQTT messaging over the internet.

No.