Do some web shopping for fans. You'll learn everything you need.
Filters are more likely to be found at hydro shops, or you can make your own, and rollitup recently wrote a tutorial on it.
Yes, exhaust fans do their job, and oscillating fans do theirs. Plants like to have air blowing on them some of the time, but never constantly.
When you remove air (exhaust) from a space, you must provide a place for fresh air to be sucked into the space. The general rule is to have twice as much area open, as you are removing. IOW if you are blowing out a 4-inch duct, then you should have at least 2 4-inch holes (or equivalent area) that are open for fresh air to go in.
You can easily see if you don't have enough area for fresh air to come in by listening to your exhaust fan motor. If it sounds like it's straining, it will go up and down in pitch a little, and you know that you need a bigger opening. If that fresh air opening is your door, then it will have to be open whenever the exhaust fan is on.