Sony MBH20 comes with Sony battery only. As per company, it offers up to 100 hours of standby time and about 7 hours of talk time.
No. Sony MBH20 do not have volume control button. You cannot increase or decrease call volume but you can use your mobile phone's call volume control button for the same.
None of the users have reported any heating problem about Sony MBH20.
Yes, You shall receive USB Cable for the same as part of the standard packaging.