Yikes! Okay, well, first problem is that your tank was never fully cycled, which is why you are seeing ammonia and nitrites now. Read these links and they should help you with understanding cycling a tank as well as the various water parameters:
https://www.livingreefs.com/cycling-tank-adding-fish-and-corals-t26452.html
https://www.livingreefs.com/water-chemistry-t31270.html
https://www.livingreefs.com/some-common-reef-chemistry-misconceptions-t41957.html
I'm also linking to an article on choosing fish that might be helpful:
https://www.livingreefs.com/choose-your-fish-t27083.html
A few things you'll need to think about to improve your tank:
(1) Rock - you really need 1-2 lbs per gallon, preferably placed on the glass in order to avoid rock slides as sand shifts and dissolves over time. You can get mostly dry rock - marco rocks and bulk reef supply have good prices, and then seed the rock with a few pounds of live rock from a local LFS. Through the course of cycling your tank the dry rock becomes live over time as the population of bacteria grows on it - that's really all that "live rock" means, though sometimes you'll get cool hitchhiker corals, sponges, or inverts on your rock as well. In saltwater, we rarely use artificial rock or decorations because they just don't provide enough surface area for bacteria to grow (as well as trying to keep the environment as natural as possible). You'll also want to stay away from things like volcanic rock and other types often used in freshwater as they leech harmful stuff back into your water. You mentioned you have lace rock - can you say more about this type of rock or post a picture? I don't really know if its safe or not because I'm not sure what type it is....
(2) Filters - in saltwater, the live rock is your primary filter. People often use a skimmer as well to help remove undissolved organic materials and waste from the water column. A skimmer creates lots of bubbles, organic matter sticks to the surface tension of the bubbles, which then rise and dump into a skimmer cup, removing the waste from the system. You can get hang-on-back skimmers or can put a skimmer in a sump system. A mechanical filter (such as a sponge, canister filter, etc) just traps the waste but it stays in the water, continues to break down, and can contribute to nitrate problems, etc. If you have any sort of mechanical filter (and you do with your hang-on-back filter and sponge), then you'll need to wash them out regularly, and rinse your sponge out a few times a week. Same is true if you have a sump system and use a sock filter or anything like that. Some people use HOB filters for flow, or convert it to a small refugium (put some chaetomorpha algae and live rock rubble in it - chaeto will help remove nitrates from the water), and occasionally run carbon a few days a month to clear the water.
(3) Clean-up crew - you need to have a clean up crew in your tank to help eat the left over food and waste your fish produce. Nassarius snails are a must - they liev in the sand and help aerate it. Turbos are good at eating algae, but you'll also want a variety of other snails as well, as the types of algae they eat vary and the smaller ones can get into the cracks in the live rock better. Many people also add hermit crabs (though they can eat your snails) and other inverts like urchins, brittle stars, etc., to help eat detritus and algae.
(4) Flow - one 550gph powerhead is not nearly enough flow. In a FOWLR (fish only) system, you should aim for at least 20x turnover for your tank size. For a 55g, that's 1100gph - double what you currently have. And more flow is almost always better - if you ever want corals, you'll want to aim for 40x turnover in your tank. And best to get a few more powerheads to add to teh one you have - that will help create more random flow in your tank and currents like the ocean - just having a single large powerhead doesn't do that.
(5) Food - I think you're feeding too much - many of us only feed every other day. Excess food can lead to nitrate and phosphate problems, and in some cases can spike ammonia and crash a tank. I'd reduce your feeding a lot - my only hesitation on this suggestion is the eel - perhaps someone who has experience iwth eels can chime in and help out with how often to feed them.
Anyhow, this is a lot of info at once, but should at least point you in the right direction. At some point I think you need to adjust your stocking - your tank is pretty small for the eel and puffer and you can really only have 1 fish for every 10g in a new tank, so you're pretty close to your limit, which is really risky in an uncycled tank. At this point, I'm not sure of the best course of action with your fish... you really need to get your tank cycled, but that will likely kill your fish in the process. Best option is to check with your LFS and see if they will board your fish for a month while you cycle your tank and get it up and running properly.
EDIT: Also, your test kit is probably okay - the brand and type are good, but sometimes they go bad even if they are new. If your LFS is getting different readings, they might be using test strips, which are notoriously inaccurate. You could pick up another test kit to test your results against. At this point I'd figure out how to get your tank cycled and then test your water daily during the process. If your kit reads 0 on ammonia the first few days, then I'd suggest picking up a few kit as your kit may have gone bad.
And finally, LFSs and their employees aren't always knowledgable about saltwater even if they are good intentioned. Some owners have been around for a long time and often recommend outdated methods; many are knowledgable about fresh water but not so much about saltwater. Many employees have never owned a tank themselves and don't fully understand things like cycling, stocking importance, etc. And finally some employees are of the mindset that those coming in and making a purchase have already done their homework and know what they are doing, so they don't ask questions and people then unknowingly often get stuff for their tanks that aren't going to survive. Many many people here have been mislead by their LFS, and it sounds to me like certainly your first LFS did that to you, and the one you have now also sounds like they are misleading you a bit with filtering recommendations. You are really best doing a lot of reading and homework on your own and then going in and asking for what you want, rather than relying on them to point you in the right direction. LivingReefs is a good place to ask a lot of those questions and to get info - no agenda on our part other than wanting to see you have a successful and healthy tank long-term!
