You do not have to remove the filter but it is recommended if you turn the nanos into a saltwater tank. The filters do gather the gunk from the water and that gunk just sits there causing nitrates which the algae loves and grows off of. If you do keep it, it is recommended to wash it out every water change in the old tank water. I also replaced the skimmer the tank came with for a hang on the back one that works better. You should not have a fishy water smell at all if you are keeping up with water changes.
If you have high salt why don't you just take a few buckets out and add fresh water??
How often are you doing water changes? I think you should be doing them weekly until you have your algae outbreak under control. What kind of water do you use? Tap water should not be used, ro
water should be. Do you have a refractometer?