I'd run both, hot and cold. Two reasons;
First, you want to always dissolve the tanning agent in hot water.
Second, the tanning baths should be between 65 and 75 degrees during tanning.
What I do is, lets say I'm mixing up a 50 gallon tanning solution. I'll fill a tank with 40 gallons of water and add the appropriate amount of salt to that. Mix it up until all the salt is dissolved. Then a take a smaller vat, put my 10 gallons of hot water in it, and add the appropriate amount of tanning agent, and mix that up intil it's all dissolved.
Then I take and add the 10 gallons of dissolved tanning agent to the 40 with the salt. The 10 gallons of hot water brings the temperature of the entire bath up to a respectable temperature.
Before or after the softener. It's half a dozen of one and six of the other, Silver~. I'd take a PH reading of the water before and after the softener and go with which ever had the lowest reading. Either way your going to have to do some buffering to the final baths to get the PH level where you want it. By starting with the lowest PH level you can from straight tap water, you'll do less buffering.