A water softener works via ion exchange. What this means is that the calcium and mahnesium ions in the water -- the ones that leave the limescale -- are exchanged for sodium ions. That is good for people, since sodium is much less likely to leave mineral deposits, and doesn't cause soap to become slick and hard to wash off. This can be bad for fish, since sodium basically is salt. As in sea salt/brackish water. You will need to research your fish and know how they will react to much higher sodium concentrations. For example, if you are keeping Rift Lake cichlids, you are probably fine, as they are quite used to high mineral content including sodium. If you are keeping discus from the mineral-free Amazonia base, they would most likely respond very poorly to increased sodium.
In every house I have seen it, the water softener goes in the line before the hot water heater, so yes, it softens both the hot and cold water. But, the water softener does not perform any other function other than the ion excahnge, taking calcium out and replacing it with magnesium. If there are other things in your water, like copper, the water softener does not affect them in any way. The water softener is not a filter, it only does the ion exchange. If you are looking for a braod filtration system, you should look into something like an RO unit, though those are going to be much more expensive.
