I thought it worked over WiFi, not Bluetooth. If it's WiFi, then your range will be however far you can go on your connection...so potentially pretty far.
I think it uses bluetooth LE (which i don't think you can turn off) to ensure you are near your your iPad or Mac then sends the signal over wifi. So the range is whatever range bluetooth LE works on.
Continuity (SMS and calls) works over WiFi.
The rest of Handoff (Safari, Mail etc.) works over Bluetooth and your devices need to be connected for this to work.
Can't speak for Mail but Safari does not need a Bluetooth connection between devices. At least, they don't need to be paired. Works fine between my iPhone and iPad and they've never been paired.
Turn Bluetooth off on one of the devices and see if it still works (I'm curious). I know that the devices don't have to be paired manually because the software has already done that and authenticated for you.