Good evening! What does these messages mean? And is it official from Apple?

1.

For that message: see this Apple  article  ☞  If you can’t turn on or sign in to iMessage or FaceTime

https://support.apple.com/en-us/HT201422

Perhaps Is your Wi-Fi network Is unavailable?

Turn on Wi-Fi network: Settings  ☞  tap on Wi-Fi  ☞  turn Wi-Fi ON

2.

The range of the mobile network is minimal. Do not wait until 'No service' appear.

See this Apple article:

If you see No Service or Searching on your iPhone or iPad

https://support.apple.com/en-us/HT201415

3.

Low Power Mode is On.

Charge the iPhone.

Wait until the battery is fully charged (100%).

The Low Power Mode will turned off automatically.