Volume level from the phone through the BT device should respond to either the volume buttons on the device or the volume on the phone (both). The problem might be that the tone to go ahead may be short in duration and the BT device may take a couple of seconds to get going, so the tone may have already sounded by the time the device is ready. If you start up the voice command thing and it triggers BT, it is probably ready before the BT device (so you could try going ahead once you hear that BT kicks in).
Sounds when something is received on the phone is called Notifications on an Android phone, not sure what it’s called on the iPhone, but probably similar. You should be able to find a section on the phone, under Sounds maybe, where you can turn notifications on or off for each thing that can receive something (texts, email accounts). If it notifies you now, without the BT device, that you have a text or email, it should notify you when using the BT device. But, the same problem as above may exist, in that the tone may be so short in duration that it has come and gone and the BT device does not pass it on in time.
On my setup, the mics are muted when BT kicks on and there is a distinctive BT starting up sound so if I don’t actually hear the tone from the phone at least I know something happened and I check my phone.