It's possible let it always connected to power

Hello, I'm trying to develop a power fail detector that send an SMS alert. I think that MKR GSM 1400 can make this task so easy, but I have two questions.

The battery connected will be charge when is connected to USB or Vin?

This is important because if the power goes down and battery starts to work, when power returns the battery should be charge completely to be ready just in case another power fail occurs.

Has it any protection circuit to cut power to battery when it is completely charge? Then is safe get the MKR GSM 1400 always connected to power?

This is important, because, I don't like the battery explode and generate a fire.

Thank you.

The 1400 does have charge protection so you will be fine.
A quick look at the schematic would say it is quite similar to the one used on the other MKR boards for which I have had zero issues despite multiple charge/discharge cycles.

I only use the USB connector so I cannot speak for the VIN without digging a little deeper into the schem.