Welcome to the great world of aquariums! Every single one of us has bloodied our nose a few times, so please, don't worry about offending anyone. You're asking the questions, and that's a fantastic start!
If I'm reading what you posted correctly, I've got a couple observations...
First, the guys at the stores need a kick in the ding-ding! They may not have technically done anything "wrong," but I want to think if they had asked about your tank, they would have suggested waiting a little while before adding more animals.
Yes, I know... Not their job... But... The better employees will want to know what's going on, and different stores have different people at different knowledge/experience levels. I'm not knocking them, but just know which person at the store you're getting advice from. We can all tell you stories - both good and not-so-good.
So keep asking here!
It sounds like a combination of three things:
First is the question of complete cycling. Test ammonia and nitrItes (nitrAtes aren't as important right now). Granted, Salifert and SeaChem have really accurate tests, but for your purposes, the API will do the job. When they both read zero, then the water is ronanly ready to being another animal in.
The second is if I'm reading correctly, there are already a few fish in there. So while the tank may have cycled already, adding a few fish will need to have all the bacteria catch up to the new work flow so that it doesn't get outpaced. I'd probably let it ride a little while before adding any more, but that's just me.
Third, which has been mentioned, anemones are weird people! They are wild cards. When you get a great one, you have a really great one! But in my experience, they can be quite fragile and finicky. I have seen entire shipments of anemones go down. I've seen one king of the tank. I've seen aquariums full of them, thriving away. Maybe next time, start with one and let it get well situated and at home.
Sounds like you're getting on the right track! You'll find lots of information here, and different hobbyists have different experiences. So ask around and pick the brains here. We're all in this together!