Can I use this capacitor for my ceiling fan?

Good evening. We're tackling a downstairs bathroom project (new ceiling, light, fan, paint, etc...) during our long holiday weekend. I've replaced numerous fixtures around the house, but haven't run into anything like this. I imagine it is fialry common, just something I haven't seen. I'm hoping someone can give me a heads up on this wiring. There are three ceiling fixtures (each with a single switch) and one wall outlet on the circuit. So each switch powers one and only one ceiling fixture, each fixture is controlled by only one switch, and the wall outlet is always hot. I tore out the existing ceiling in the bathroom, so I have the wiring exposed to the old fixtures. Here is where I am confused. Each fixture has the romex with ground going to it's switch. The fixtures are also connected to each other (I haven't tested each wire, so I'm going based only on color now): Vanity fixture- 1) Romex from switch to fixture. 2) Romex from fixture out to ventilation fan fixture. The two hot wires are connected just to each other. One neutral is connected to the fixture hot; and one neutral is connected to the fixture netural. Grounds are connected to each other. Venitlation fan- 1) Romex from switch to fan. 2) Romex in from vanity fixture. 3) Romex out to light fixture in adjacent hall. The three hot wires are connected just to each other. The three ground wires are connected to each other. The neutral wire from the light switch is connected to the fan's hot wire. The other two neutral wires are connected to the fan's neutral. Adjacent hall fixture- I haven't checked it since it is in a different room with the ceiling intact. My gut tells me if I take it down it will be wired similar to the vanity light... with Romex to the light switch and the Romex in from the ventilation fan. I guess I'm confused as to why the fixtures are connected to each other since they each have a dedicated switch. I've seen it where a line of lights is connected by a single switch, so power has to go from one to the next and so on. Why would these be connected to each other? When I hang the new fixtures should I wire it the same way? That is a ton if wiring to squeeze into the ventilation fan since it is already a dual fan/light that I'll be running off the single switch. Thanks in advance for any input. And have a great holiday weekend!

I am in the process of trying to install a ceiling fan in my son's room and need some advice. The fan has a wall control and the instructions state "Warning: Hook up in 'series only' Do not connect the hot and neutral wires of electric circuit to the wall control." The box on the wall just has one black and one white wire, which I assume are the hot and neutral wires. But, the fan is replacing a light that had a dimmer, so why can't I just wire the wall control for the fan the same way as the dimmer? Thanks in advance for any help. Sean