i am "cute" busy ..........I think you wanted to say : I am "quite' busy because the pronounse of "cute" is simillar as "quite" but they are much difference in the meaning !!!! "cute" means 'hollwa" in ( pretty busy) place ....... that is ok i think this sentence is quite right !!!! the "pretty" in this sentence means "very" not means "beautiful" at this time and you can use "pretty" to substitude the first sentece as "quite" :D

"Cute" does not work in either sentence, but "pretty" does. You can say, "I am pretty busy." You can say, "...in a pretty busy place".

I am quite busy. in a pretty busy place.

No this is not correct. There are words in phrases, which we can't replace with others, even though the other words will mean the same.