You could definitely get by without a HOB filter and just water changes. However, you may wind up struggling with nitrates in the beginning stages of the tanks life, and after detritus starts to build up.
With that in mind, i would strongly suggest a HOB filter, like mentioned above. You could go with the AC50... OR, you could upgrade to the AC70, and mod it and turn it into a fuge to hold your heater and some chaeto, which will definitely bring down both your nitrates and phosphates. Also, the good thing about getting the AC70 instead of the AC50, is the added flow. I know many people with 10G AGA tanks that say there's even too much flow in there tank with the AC70. I'm not sure what your aiming for as far as coral, but if you plan on doing SPS the AC70 would be a good choice. Also, if needed, for added flow you could use a Koralia Nano.
As for how long after you can add coral and your feather dusters, i would say any time after your cycle is complete. The feather duster should be fine just going in there, but i would wait a couple days before you put any of your prized corals in there. Just to see if everything is going alright. Plus, by the time you began adding corals to the tank you would want your Nitrates and Phosphates to be as low as possible. Plan on adding any fish or anything that you will feed the tank for? My only concern is, if you add the feather duster in their alone, it may not have food. In aquariums, they usually feed of of floating particles in the water, and mostly left over food particles floating around. If you have nothing in there to feed, it may starve. They are not photosynthetic.
Also, are you putting any live rock or sand from your main tank into the pico? If you put live rock from your main tank directly into your pico you may be able to skip the cycling process all together. It would be like adding fully cured live rock. You MIGHT see a small cycle though, so if you do go this route, i would wait a couple days and test the water consistently. If everything reads zero after a few days, you should be good to add your corals. I see many people with pre-existing aquariums do this when they start new tanks. Mostly smaller aquariums just like what you are doing.
Here's a link to modding an AC70 into a mini refugium:
Modding an AC70 HOB Filter