It looks like tax rates range from around 2.5% - 2.9% of appraised value, depending on which city and which school district the home is located in. In those zips there are the cities of San Antonio, Alamo Heights, Terrell Hills, and Olmos Park. There's San Antonio ISD, Alamo Heights ISD and Northeast ISD in those areas as well. Appraised value tends to run below selling prices based on what I've seen. Based on a couple of studies I did a number of years ago, I'd guess about 15% below on average.
Median rents will vary between areas. Based on current asking prices and all types of properties (homes, condos. etc):
Alamo Heights ISD is approximately $.89/sf (40 rentals)
Northeast ISD is approximately $.80/sf (11 rentals)
San Antonio ISD is approximately $81/sf (46 rentals)
Of course these will vary depending on type of rental, size, condition, etc.