I bought a starter pack of Panasonic Eneloop rechargeable NiMH batteries about 4 years ago and I've been using them in all my wireless peripherals since (mice and controllers). I highly recommend them as long as your peripherals are fine taking 1.2 V instead of 1.5 V (which is most of them).
Any regular xb1 controller since 2016 has bluetooth. The old elite controller doesn't have bluetooth but the new 'elite 2" does.
If you're not sure, look at the top section surrounding the xbox button. If it's the same color/texture as the main body, it's new. If it's the same color as the bumpers, it's old.
Also remember that any micro-USB cable (not USB-C) will also allow you to connect the controller to Windows 10, without depleting your batteries.