Hi Kolooko,
I believe the correct phrase would be as follows:
"He is of normal height."
The use of the word "of" being the most common or correct choice in this case. However, there is a problem with the use of the word "normal" in this statement. What is "normal" when we are talking about something like height?
"He is of average height" is probably the better way to say that someone is neither tall nor short.
Again, to answer your question, I believe "of" is the word for which you were looking.
Examples:
He is of Japanese descent.
She is of average weight.
They are of the same family.
...and a fun, popular expression.
Birds of a feather flock together.
As for your additional questions...
"He is normally high" would actually mean "he is usually under the influence of drugs".
"He is average in height" would work instead of "normal".
I hope this is helpful to you.