Nanook's instructions are correct - generically at least - I don't have a Neptune controller so I can't comment on that particular unit.
It's true that the reading you get does depend on the tempreature of the liquid. However, there's not much difference and for our purposes it's perfectly okay to ignore this difference.
Many of the calibration solutions only read the correct value at 25C. Again, the values they read at other tempreatures are pretty much the same so this isn't worth worrying about too much.
A probe that has been allowed to dry out will need time to rehydrate. A couple of hours is usually enough. Best to leave the probe in tank water 24/7. It's essential that you do not store probes in RO water or similar for long periods of time.
Probes need cleaning from time to time - you can buy a solution for this. It's essential that you recalibrate the probe after doing this.
Probes do eventually wear out too. My last one went for four years before it started getting really sluggish and became impossible to recalibrate.
As for the calcium reactor effluent that is very low for most types of media. As it happens, I have to run my reactor as low as yours to get adequate alkalinity in the effluent - I use the AquaMedic reactor with the standard media it comes with. At a pH of 6.5 I basically get tank water out of it. Getting the pH that low just requires adding heaps of CO2 - I imagine this is getting close to saturation - but I don't know for sure. The statements I've seen on not being able to get salt water pH under 6.5 are wrong IMO anyway
These days my probe is used to control the pH inside my reactor. If I was not doing that it would be in a cupboard gathering dust i.e. I happen to have a controler and once decided I may as well make use of it. I guess pH is one of thoe things that's always about right if everything else is about right too. The only time I test the tank itself is if I really can't think of anything else to do - that happens rarely