The temperature shown does come from the Weather App. Small bits of data on the watch face are known as complications.
Tapping the temperature on the watch face will open the Weather app thus revealing the source of the data. The watch gets the data from your iPhone which, in turn, uses location and data from the Internet.
You may find more useful information in the Apple Watch User Guide.
I imagine the WATCH will keep as much data as its onboard storage will allow, which is several gigabytes. So, caching Health data really shouldn’t be an issue for most intents and purposes.
Since the title of the question is about the disadvantages of Airplane mode, know that placing your WATCH in airplane mode will preclude you from connecting to the Internet. Therefore, any feature that requires an active Internet connection will not work. Siri comes to mind.
Finally, since the Bluetooth antenna is also disabled, you will be unable to connect a set of Bluetooth headphones, or an external heart rate monitor. Any app that requires an iPhone connection will not work either, meaning all watchOS 1 apps, as well as many, if not most, watchOS 2 apps. GPS via your iPhone will also cease to function.
Enabling Airplane mode thus disables most of the features of your WATCH. On the other hand, there aren’t too many advantages to enabling it. You might see some gains in battery life, though most of that will come from not being able to use your WATCH for a whole lot of things. But if you weren’t using your WATCH a lot to begin with (outside of activity tracking), then battery life is more than adequate to last you an entire day.
Even if you are on a marathon hiking trip, you will defeat much of the purpose of tracking that workout anyway, since the lack of GPS will result in less accurate distance measurements. In that situation, it would be preferable to turn on Power Saving Mode under Apple Watch → Workout.
Best Answer
To troubleshoot, I would follow the restore steps:
It mentions that bluetooth should be on on the phone and you should be connected to WiFi so if you can turn both of those on on both devices (might be hard on watch unless it prompts you) but if you can get the same WiFi router as the last time it booted, it might reconnect automatically.
Once that’s done, you might end up down the “watch won’t restart” article which has a few more things to try before you contact Apple Support. Even though force restarting the watch isn’t ideal, you might try that as well one time once WiFi and Bluetooth are enabled to the max extent you can manage.