The only shop that sells potions is near the Windswept Jail (where you get the Boomerang). To get there, you must open a cave to the north of Hyrule Castle (requires the arrows) go through it, switch the time to twilight, go through the cave on the far side and south.
I have no idea what's causing some people to find those chests empty. I'll have to do some investigation.
The NPC who wants the Bug Net is in the same area as the one who gives it to you, except at night. There's a bridge to the west of the dungeon which can be crossed then and he's on the far side.
The bomb and arrow shop in the castle should open when you have both the bomb and arrow and talk to the guy who tells you about it.
That script... it does this at times for reasons I don't understand. I quite literally have to be pixel-perfect about placing those FFC's.
The feather is in the Volcanic Fissure. You won't be able to get there until you have the Magic Cape. (From the dungeon in the sky.) Once you do, cross those spikes near the Gerudo Stronghold to reach it.
I tried to do that when I just got the bombs and arrows and talked to the guy and he didn't unlock his shop. Thanks for the tip about the net, it worked!