Is there any relationship between the sound emitted from a rocket engine and the amount of thrust it gets at surface temperature and pressure?
The short answer is yes! I've heard the relationship referred to as "acoustic efficiency" (i.e. how much of the power of the rocket essentially makes sound).
I don't have a lot of experience in the relevant areas, but this NTRS paper has some references that might let you explore the question. (Bonus for space archaeologists: as late as 1999 (which I infer from the reference dates and the URL) NASA was producing electronic copies of papers by printing them out (probably from Word) and re-scanning them...skewed.)