There would be power there because the "neutral" is the path electricity mostly used to return to its source. All electricity has to return to its source somehow.
An ordinary light switch does NOT have any neutral -- it only switches the live wire. Some installers use the white wire as a "switched" wire to the light fixture, where the power cable went to the light and a separate cable went to the switch. Technically, the white wire should have been marked with red or black tape at each end, to signify it is not neutral - it is either live or off. In that case, you would not be able to use a PIR switch because there is no neutral, which is necessary to provide power to the electronic circuit in the PIR. You would need to rewire the switch with a new power cable from the branch circuit. The you have the live and neutral wires in the outlet box, connect them to the PIR, and run the pigtailed neutral and the switched live (and pigtailed ground) to the light fixture (where you would disconnect the other power cable completely and cap the ends separately).
That led is only showing that there is power to the switch and/or the light and fan fixture. You would have to check the power at the fixture to determine if it is actually in the switched feed or the fan/light assembly.
A fluorescent light fixture is designed to operate on an AC voltage supply. To have the fixture operate off of a DC supply a power inverter would have to be installed. The power inverter would then change the DC to AC for the fixture to operate. These power inverters are reasonably priced these days and can be bought at most DIY stores.
The switch you are referring to is called a three way switch operating one or a string of devices from two separate locations. You have to either parallel it at the existing light or find out which end the switch leg and associated neutral is on. Sometimes the electrician will feed the fixture with the neutral and use travellers between switches with 3 wire circuits so you first need to know how it has been wired. It is legal to feed the switches and lamps from any of the locations. Unless you are tapping into the light fixture itself, I would have a professional look at it.
The neutral wire is a return wire for the current in an electrical circuit. Do not confuse for the ground wire which is also a return wire but is used in the event the connected appliance shorts to protect the user from electrical shock. The neutral is actually very similar to the ground, though. In a residence the neutral comes from the power plant, whereas the ground comes from a ground rod below the meter. In most older homes the ground and neutral were connected to the same bar in the breaker box. In newer homes they now have separate bars. Here is something interesting about the neutral wire. If you are testing a live circuit using a static checker, the neutral will not show a charge, only the hot wire will. However, if you had a circuit controlling some device (maybe a light fixture) and the light fixture was in the on position, but the neutral was cut you would notice the two wires slightly sparking when you touch them together. If you were to complete this circuit with your body you will get shocked or electrocuted. If the device were in the off position you would be safe, but don't take any chances when working with electricity. Turn off the power first.
On these types of installations an electrician would use a three wire cable set from the switch box to the fixture's junction box. At the fixture's junction box both the fan neutral and the light neutral would be wire nutted together with the incoming neutral from the switch boxes. From the three wire, the white wire would go to the fan and light's neutral wires. The Black would come from the bottom of one of the switches and go to the fan's motor lead. The red wire would come from the bottom of the other switch and go to the light fixture's lead. It can be done with two runs of two wire to the fixture. You will need a two gang switch box for this project to hold the two separate switches. You could use a single gang light switch box if you can still find where duplex switches being sold. A duplex switch is one that has the configuration of a duplex receptacle with the switches being one on top of the other. The cover plate is the same one used for duplex receptacles. To answer this question fully more information is needed. See discuss in the left margin.
Get the right lamp or get the right fixture. Do not attempt to modify the fixture to fit the lamp.
You would have to provide more info to be certain what your problem is. Typically a black and white wire would connect to the two wires on the light fixture. The fixture should also be controlled by a switch that "breaks" the black (Hot Wire) to turn off the fixture. If you have a volt meter you can see if there is 120Volts between the black and white wires with switch on. If there is a single white wire going to fixture (Neutral), where is the "black wires tied" to. Maybe one of the black wires has broken off the fixture.
One. A typical "two way" ON-OFF light switch is a Single Pole Single Throw (SPST) switch with two terminals. It is wired in series with the fixture: that is, the power circuit is routed from the breaker to one side of the switch. The other side of the switch is connected to the "phase" or "power" terminal of the fixture. In a 115VAC single phase circuit, the other terminal of the fixture is connected to neutral. A "three way" switch circuit consists of two switches in different locations, each of which is a Single Pole Double Throw (SPDT) switch with three terminals. A typical use would have one switch at the top and the other at the bottom of a set of stairs. For the sake of clarity, we can label the "common" terminal on a SPDT switch the "C" terminal. This terminal is internally connected to one of the other two terminals, call them "A" and "B", depending of the position of the switch. A three way switch is connected with the circuit breaker connected to the "C" terminal on either switch. Conductors connect the "A" terminal on one switch the the "A" terminal on the other switch and the "B" terminal on one switch the the "B" terminal on the other switch. The remaining "C" terminal is connected to the "phase" or "power" terminal of the fixture. The other terminal of the fixture is connected to neutral. Note that, in either case, there is no neutral wire connected to the switch. Ground conductors do connect to the chassis of the switch box and the fixture, but they do not carry current and do not affect the operation of the circuit.
If you don't know how to install light fixture, I don't recommend that you do it by yourself. The installation theory is actually very simple - all you need to do is attach fixture on the ceiling (or any other place) and then connect the electrical wires (usually there are two of them). After that it's good to put the light cover over the fixture to improve the appearance.
It would be a fixture.
The headlight fixture probably contains moisture on the inside. When the headlights are "on", the moisture clouds the light beam, diffusing the light.