If the Hospital No 11 was where the University of Nashville was located, that area has been revised by Davidson County (Metro) government buildings. University St is not there now. The Howard Office Bldg, Davidson Co Motor Vehicles, Metro Codes, Davidson Co Fire Marshal and Nashville Children's Theatre plus their parking are located at 700, 730 and 800 2nd Ave South. The complex is between 2nd Ave South and Hermitage (1st Ave which abuts the Cumberland River) and Middleton St. and Lindsley Ave. I think University St may have been about where the Nashville Children's Theatre is located today.
Don't know if you know about Lower Broadway where the honky tonks are so popular these days. The area you want is south of Lower Broadway near I24.
I don't know where the soldiers who died of smallpox were buried. The National (Federal) Cemetery is located in a suburb named Madison on Gallatin Rd and Briley Parkway.
Seems I read somewhere that Yankee soldiers were buried in the Old City Cemetery before the National Cemetery was opened. Some Confederate soldiers were laid to rest at Mt. Olivet Cemetery on Lebanon Rd.
Hope this helps.