Can MBP 13 mid 2012 send audio and video through thunderbolt to HDMI?
As far as I know, thunderbolt to HDMI by default provides audio.  Now, whether your adapter supports it, well that's another question.

I have used this device from Amazon Basics to supply HDMI to a TV (including audio) from a MBP and a Microsoft Surface Pro 3.  It worked with zero issues.