There's no power going to the radio when you shut off the FJ.
So yeah, that's normal.

 

Normal, same deal on my Taco.

 

For some reason, when you connect the Bluetooth to either my or my wife's phones, it doesn't stay connected if the engine is turned off, even for short amounts of time (I'm talking 2-3 minutes).

Is this something normal to the stereo?

Just to clarify, since the two posters above me are correct. Is the radio powered on -- the faceplate lights are on, you are listening to it etc -- when the engine is off? You are using the radio, but for some reason the bluetooth connection goes away?

The radio is not powered when the engine is off, unless you have the key in "accessory" mode (at which point it's powered as all the other accessories are). If the radio is on but the bluetooth isn't working, it's possible you have an aftermarket bluetooth module that isn't powered when the engine is off.

 

My radios have a constant hot and a switched hot feeding them.