if its just carbon i would leave it in. if it has a filter pad with it, this needs to be cleaned every other day or it will cause problems, best without it.
the "white strings" appear to be mini feather duster tubes. these are worms that filter feed from the water and seem to come and go depending on food source.
actinic lighting isnt required for corals but makes them appear more pleasant in coloration. is this going to eventually be a reef? if its going to be fish only, a 14k bulb would be just fine.
get yourself a heater. i suggest jager.
welcome to the forum.