“Could this continual weekly water change damage the heater in any way?”
No, I cannot really see how it could affect the heater, as long as, they are adding the appropriate amount of balancing chemicals (calcium, alkalinity, ph) each time they drain and refill the spa. If they do not add the balancing chemicals, it will affect the heater.
“This is a relatively small spa (325 gallons) and it took about 18 hours for it to heat up from the fill temp of probably around 60 degrees.”
The spa is probably running 110V. It takes some time to heat up using 110V.
”Also, what's the deal with the control panel not displaying the 'actual' temp as GMAN questioned in his thread? Is this common for control panels to only show the 'set' temp?”
Hotspring is just different. Most manufacturers show both the set temp and the current temp. I don’t think it really matters.