Can my baby get a rash on his back from public baby-changing tables?

No. It is very unlikely for an older infant to get a rash from any kind of changing table. If desired, a parent can wipe down a changing table with some diaper wipes, then dry it with some paper towels, before putting the baby on it.
Yes. Yes, but a few simple procedures can minimize the risk. You can wipe down the changing table with a baby wipe or cleansing cloth, and then place a changing pad, small towel or blanket on the table before you lay your child down to change their diaper.