Good to know where volume settings are hidden, but nope. According to others the microphone does not work too...
EDIT: Bluetooth works in another headset device. Just tested it.
EDIT2: OK, reset of the phone, tried CM11, too. No dice. Seems to be a hardware problem...
Samsung Galaxy Note 3 | SM-N9005 | Crash ROM v.17, Bootloader + MODEM XXUGNG1, updated Adreno drivers to Aug. 7th Version, Ultimate Kernel v.65 (Thx Mcaserg for your Kernel)
Samsung Gear | SM-V700 | TizenMod v3.0 by Skin1980 | Pretends to be a SM-R380 for moar Apps
Lenovo ThinkPad Tablet | 18382DG | JellyBean 4.1.2 CyanogenMod by Koshu, rooted with Debugfs automated root from sparkym3
Samsung Galaxy Note | GT-N7000 | CM-11 NIGHTLY + PA GAPPS STOCK Pack