Is it true that NEXP = EXP when the input binary strings have size O(n)?
"When the input has size $O(n)$." Unless otherwise stated, $n$&nbsp;is defined to be the length of the input. So, generally speaking, the input length is just&nbsp;$n$, and your question is just


  Is EXP equal to NEXP?


We don't know, but we suspect not.  Nondeterminism seems to add extra power that we don't know how to simulate deterministically without requiring exponential time.


  If an language is in NEXP, then we can list all input possible input strings and then use exponential time to check each one


That doesn't make sense. In a computational decision problem, you are given an input and you must determine whether the answer is "yes" or "no" for that specific input. Enumerating other possible inputs doesn't seem helpful: if you're asked a question, your job is to answer that question, not list off all the other questions you could have been asked instead.