Q: Bluetooth text messages

It is also car dependent. You can display text messages from at least some models of iPhones with some models of cars. You may need to contact your dealer or see if your car manufacturer has more information on their website.

As far as I know, only some data between Apple devices can be exchanged via Bluetooth, for example contacts. No other Bluetooth profiles are supported than handsfree, headphones or keyboards, playing music etc.

iOS: Supported Bluetooth profiles - Apple Support

I'm not sure what profile it is included in but my car does support displaying incoming text messages from my iPhone 5s.

Edit: it is the MAP profile (iPhone 4 and later) in iOS: Supported Bluetooth profiles - Apple Support

With my car, on the phone I go to Settings > Bluetooth and tap the blue i in a circle beside the vehicle name to get to the vehicle-specific Bluetooth options. You've likely already tried that, but just in case.