The shrimp method tends to put A LOT of ammonia into the water in the first few days, which is why it takes so long for the tank to cycle afterwards, it takes a long time for bacteria to build up to process all that ammonia. Again, nothing wrong with that method, been used for decades and it does work, I just think there are better methods today but if someone were to use the shrimp method that is fine as well as long as you have patience (which is a requirement for this hobby anyway so it is a good lesson to learn early). ok... so the live rock be the sole ammonia source?
Purigen is like a synthetic carbon that you can recharge and reuse. Chemi-pure is just average carbon, I would not recommend Chemi-pure today because Rox carbon is 10x better and cheaper. Rox Carbon has been shown to remove up to 90% of DOCs out of a tank (Dissolved organic compounds) where as a skimmer can only remove around 30% so carbon is much better than a skimmer is in that regards but you have to replace carbon often so at some point a skimmer becomes cheaper. Also a skimmer will help out if you overfeed your tank by removing large food particles, but to me with a coral tank, I want those to stay in the tank so I do not run a skimmer, just an algae turf scrubber, so when the food does start to break down into ammonia the scrubber/algae will remove the ammonia from the water. A skimmer also helps out with oxygenation of the water where as carbon does nothing in that regard. So skimmers do serve other purposes that are beneficial, but also do negatives like remove to much good stuff out of the water.
i'm not doing a skimmer since it's only ten gallons. so i need carbon? does it have to be those types, or can it just be the stuff that comes with the filter?
If your using dispenser water, buy a TDS meter to check the quality of the water. You can get water that is worse than tap out of some of them.