I love hearing stories like yours. Too often students from public rural and city high schools are overlooked by adcoms who focus on magnet and elite private high schools.
-yes, Vandy will meet your finacial need
-yes, you will fit in
-yes, you will likely be a bit behind your peers from the top magnet and private high schools. First semester take a lighter load to adjust to the academic rigor at Vandy then you should be fine and can ramp it up in the spring.