"I really was." "Really think you." "Really don't mind."  Are these three correct?

Everything depends on context.

"You weren't there last night."
"Yes, I really was". Perfectly correct.


Really think you-- makes no sense at all- did you mean "Thank you"? 

"I appreciate you taking me home."
"I really don't mind."  Very informally- "Really don't mind". 

They are all correct if you use them in the proper context:

"I really was going to try the bungee jump if it hadn't been raining that day."
"I really think you need to give you studies more effort, or you are going to fail the year."
"I really don't mind if you want to use my back yard for your daughter's birthday party, just make sure you clean up any mess."

The first and third could be correct, but the second makes no sense.

you can use "really" to mean the broad equivalent of lots, very, to a great extent.  More than just average, closer to extremely but not quite there. I am really hungry.

A word of emphasis to mean somewhere between some and a lot in intensity.  It can also be used to mean "truly".  Sometimes we use it to mean both at the same time, making it somewhat vague as to its exact meaning, and we do it on purpose: I really do like you.  Does the person mean truly, or a lot, or both?

The first and the third ones were correct. The second one was wrong.