is it possible to have this entry box automatically start with the letter "N" when you click the box to make a manual data entry...
Yes. Create the Government ID field and double click or click Options. On the Auto-Enter tab go down to the Data box and enter N (no quotes). This will enter N for all students, not just international students. If you have a way of determining international students vs local students you could instead use the Calculated value. Click Specify... and create a calculation; something like:
If ( StudentType = "International" ; "N" ; "" )
2. If the first question is possible, is there a way to prevent data entry IF the N######## that was entered already exists in a previous record for another student?
Yes. Again, double click the Government ID field (or click Options...). In the Validation tab check Unique value.