This can all be fixed and it will just take a little time. All great advise from above so let's try and simplify things and put your next task/s in order.
Local Fish Store (LFS)
Water Change (WC)
Display Tank (DT)
Live Rock (LR)
In My Opinion (IMO)
1. Take a deep breath and relax
2. if the aptasia is REALLY bugging you and you HAVE to attend to it be glad most of it is on the glass. Simple to remove without adding juice. Save that for later on the ones that develop at the base of your live rock (LR) that you just don't want to tear down. When you "F" with aptasia it releases spores into the tank that become more aptasia. SO for the glass pests....grab a bucket, start a siphon like you would for a water change (WC) and put it right over the aptasia and use the hose to scrap it off the tank. Any spores will go right into the waste bucket. So that takes care of the glass pests. If the hose won't work, razor blade with the hose right up to it so all gets sucked up.
3. Aside from all of the main cycling issues; ammonia, nitrates and nitrites are what is killing your corals currently. Stop adding anything to the tank except your fresh salt mix used for water changes (WC).
Your salt mix will have enough of all that's good to keep your corals happy because you will be doing them every other day until your levels stabilize. (sorry to be the bearer of the bad news)
4. Here is your schedule and just a guide but I'm sure you'll get the gist.
Day 1 test water parameters then - 50-60% WC This will give you a baseline of where you are currently. Write it down!
Day 2 Do nothing but relax
Day 3 Test then - Another 40-50% WC
Day 4 Relax
Day 5 - Test then - 30-40% WC
Get it? LOL
You can alter the % of WC after day 3 to suit the needs of the tank. If your water parameters are looking much better then the % of WC doesn't need to be as large but is still needed until your levels come way down to acceptable levels.
A few things to remember going forward:
1st. and formost....your LFS is on business to make money. Also some....not all LFS owners would much rather sell you snake oil because they don't know any better. Nothing against LFS owners but some can give out less than perfect advice.
2. If you are unsure of what to buy....check here first. You'll save a TON of dough by doing so.
3. ANother BIGGIE - corals in general do NOT like drastic swings in water conditions. Too much of what may be needed is better dosed over time than all at once, despite what the manufacturer recommends on the bottle. Those recommendations are a guide as to how much solution will raise the parameter of what your dosing in "X" amount of gallons. It doesn't mean you should dump it all in at once.
4. Test kits that are easy to read are the Hanna Checkers period the end. They cost a little more up front but IMO well worth the digital read out. The only other test kit that Hanna doesn't make that you will need is for magnesium and for the most part they are all a titration test and I'd recommend Red Sea. But, any other brand name will do just fine.
5. When in doubt - Ask questions on the forum
I hope this helps...
Keep us posted!