It must be something with either the 630 phones, or you guys are trying to use Apple ear buds on your Windows Phones, which has known issues.
I've had the Lumia 920 and 1020, and even a 520 Go Phone, and not had the issues you guys report, even using cheap $5 ear buds.
I have an Oontz Bluetooth portable speaker, a Blaupunkt Toronto Bluetooth car stereo receiver in my Jeep, and an Amphony Bluetooth receiver/amplifier that power my Polk speakers on the back patio. All my Windows Phone play nice with these devices without issue. I can listen to music on all. I can send or receive calls and message alerts in the car or on the Oontz portable speaker.
I work in IT professionally. I have an office full of Windows Phone users, and they are by far the best phones we've used. Our company trucks also have Ford Sync, and the phones play nice with those as well.
One other thing with Bluetooth... If you are having trouble with connection to a specific device, select the checkbox under Bluetooth "advanced" options that says "Use an alternate Bluetooth audio connection for Speech.
And like Aaron said, you'd be better off buying a used Lumia 920 than a new 630. The 830 will be an upper mid-level phone. The specs are actually quite nice, and it is important to remember that the Windows Phone OS does not need the same level of hardware horsepower as iOS and Android do. My 2 year old Lumia 920 still bests the iPhone 5s in most tasks when it comes to speed and fluidity.