1) What is a correct way to say and write when you are talking about the person's height (5'5'')

five feet five inches or five foot five inches?

2) What is the abbreviation for inch? What is the correct way to write 5'5'' - 5 ft 5 in?

Thank you
For no. 2, in. is the abbreviation for inch. You could write either 5'5", or 5 ft. 5 in.

I prefer five feet five inches, but I hear five foot five inches probably more than I hear feet, in common usage. To me, it isn't logical to use the plural for inches and the singular form of foot, so, my opinion is that feet is correct. But, because the usage of the other is so ubiquitous, I'm sure there are those that will argue that it is now standard.
Thank you Sam.