hehe it's normal to worry
The fins, are the usual droopy fins of veil tail (most common) betta. Some bettas also tend to "rest" because they are tired, lazy, etc
I've had a lazy betta o_O If the temperature of the water is lower than 77, try raising it (you have a heater right?) if you can to maybe 78.
Not a silly question
only silly question, is a question unasked. It is normal for him to swim that way, when there is a filter on. Seeing my Dally who has no filter swim, vs Shadow who has a filter, Shadow tends to tip his nose up a little, and even swims forward with little spurts (though the filter is on low).