Wow, sounds like a nice deal. I'll comment on what I feel confident about and let other people comment on the other parts of your question, like the lighting. My guess is that more lighting would be beneficial to the anemone especially, but if it's been doing OK under those lights, maybe not.
As far as water parameters, if your cyclae is complete, which it sounds like it is, you should test for salinity, pH, alkalinity, nitrates, and maybe calcium. Usually, the test kit will tell you what the normal range is. pH should be between 8.0 and 8.3 if at all possible. Specific Gravity (a way to measure salinity) should be close to 1.026. Nitrates should be as close to 0 as possible, but not more than 15 under any circumstances.
As far as hair algae, if you manage to get rid of it, let me know, because I've struiggled with it a lot. The only thing I know to do is to use your skimmer as much as possible and do water changes to get the nitrates down to 0. Also, you can test for phosphates. If you have phosphates of any level in your water, it's promoting algae growth. Do large water changes with RO/DI water if this is a problem. Test for this and try to get all the nutrients out of the tank that are feeding the hair algae as soon as possible.