Prime is treating the chloramine and / or chlorine. The water is likely safe for fish as far as that goes since it sounds like you have used it for freshwater tanks. There's 2 things going on with the water that you aren't used to coming from freshwater:
1) As your tank grows and you want to add coral, some other things in the tap water can be problematic, like copper. Some of those can soak into the rocks now and make an issue later on. It's stuff that doesn't hurt fish or people but coral won't tolerate.
2) Reef tank lights mimic the sun, so algae can have a field day if there is fertilizer in the water. That can come from food and fish waste, and also impurities tap water.
For a 10g I wouldn't buy a rodi either. The Walmart distilled water with a purple cap is the go to for many nano (small tank) reefers. A few water changes will bring the alk down but you need to be careful when you change more than 20% of the water in the tank that the new water is the same temp and salinity. In a little tank like this it's easy to shock the animals and a couple degrees difference is hard on them.
You have another issue here tho. It sounds like you have a lemon chromis, they need a bigger tank. Your stocking options are limited in a 10g because of the fish's territory, personal space, waste production, and natural swimming habits. As opposed to a lot of fw fish reef fish generally come from the wild and they need more space generally to thrive in captivity.
Honestly in your shoes I'd return the nem and the fish, get the tank on track and then start over with appropriate pets. Here's some fish that are happy in a little tank, and down the road when things are settled you can def get a nem. This website has good info about what fish need, and you can show it to the wife on your phone in the store or whatev so you aren't the bad guy who won't get the critter
http://m.liveaquaria.com/product/aquarium-fish-supplies.cfm?c=15+2124
PS you can't just boil the water, you'd have to capture and cool the steam leaving behind the contaminates. Walmart water is like $1 gallon. Don't get the Ro or drinking water as those will have minerals that you are trying to avoid, you need distilled