Being that the carpet has been in there two weeks and your algaecide product has as well, I might be inclined to look on that as at least a partial suspect. Even though considered an organic product, any item claiming to be safe should be treated with greatest sceptiscism.
Aside from that your water specs are quite poor for sensitive inverts as these and most likely the root cause. You should really do several large water changes over the next few days with well aged (24 hrs) and aerated
SW, heated to the same as the tank. That should help with the low
alk/pH as well as lowering the nitrite and nitrates. I would seriously suggest removing the Algone as well. It will not reduce
PO4 either, it simpley binds nitrates and supposedly lowers the available nutrient in the system. Judging by the time it's been there and your nitrate levels, it's not working.
Higher nitrate (above 10ppm) is quite bad for more sensitive inverts but the presence of the nitrite is even worse.
NO2 should be undetectable at all times. How often do you feed the fish and how much/what?
The anemone attaching and the tighter mouth is a good sign though. Do not try to feed it any further either. It's not necessary given the lighting you have unless it ends up catching nothing with general fish feedings. Even then I would urge against it.
If the Emporer
HOB filter is the main means of filtration you need to be sure the media is cleaned quite often to prevent organic build up. Is this mainly an
FO system with some inverts or does it have
LR?
You will also need to make a serious choice in what you will be keeping in the tank. The Picasso will begin to look upon the anemone as lunch sooner or later. They are no where near inverts safe at all.
Cheers
Steve