I have not used that particular module, but a few using the CSR8645.
You are correct in that Vbatt is 'power' (often a lipo battery 3.2 - 4.2V) but usually these modules have a 'Vreg_EN' pin which enables the on-board voltage regulators and powers it up.
In your case it may will be the MFB pin - multi-function button. When you apply power using Vbatt, and GND, check the '1.8V' pin for.. well 1.8V. Try pulling the MFB pin high by tying it to the Vbatt with a 10k resistor.
The IO voltage on these is probably 1.8V so you should use the 1.8V line for your buttons, but Vbatt will do as long as you have series resistors to limit current. And in the case of the MFB it has to be Vbatt anyway as that enables the regulator.
Audio out is SPK_LP/SPK_LN for left, and SPK_RP/SPK_RN for right. It has differential output, so a + and - for each speaker, which means you can't just connect the negative outputs to ground. The reason for the differential connection is that there is often a lot of noise coming from the rest of the chip (power management, CPU core, RF parts) which bleeds to the outputs. By using differential connection that noise can be cancelled out, but it does make it more difficult if you want to have an output with a common ground - often requires either an opamp difference amplifier for each output, or a headphone amplifier chip with differential input (PAM8908, MAX9722 etc..).