After some experimentation, and expanding on information in the comments:

Short answer

1. Because of bug.
2. But it doesn't really matter.

Long answer

[The problem]

This happens consistently in my Sanctuary settlement too, the only settlement where I have spotlights installed. I have two spotlight turrets; each is on level ground, and each is right next to its own dedicated generator.

The problem appears to be triggered by the passage of time rather than fast travel itself. For example, a very short fast travel (e.g., to Vault 111 and back) leaves the spotlights on, but a little further (e.g., to Starlight Drive In and back) and they'll turn off.

But most importantly, the problem is only cosmetic.

[The experiment]

Wait until a spotlight turret is in its darkened state, then:

1. Note that it's still moving; this strongly suggests that the game still considers it to be active.
2. Go into Workshop mode and notice your settlement's defense number.
3. Pick the turret up and set it back down. It should be lit now.
4. Notice your settlement's defense number is the same whether or not the turret is lit, so long as it's moving.

Then, to be extra sure:

1. Fast travel a good distance, for example, from Sanctuary to Starlight Drive In.
2. Return to your base. The spotlight should be dark.
3. Check your settlement's defense number. It's the same.

[So...]

While a little lighting in your settlement would be convenient—not to mention darned pretty—this appears to be a purely cosmetic bug.

You can sleep soundly knowing that those turrets, even when dark, are scaring away bad guys. And you can also sleep soundly because those turrets won't keep shining in your window. (Until Bethesda fixes this bug.)

The power armor radiation gauge simply gives you an idea of what radiation is present in the surrounding environment.

It's important to be aware that, while certain things like power armor and hazmat suits greatly reduce radiation damage, nothing will completely negate all rads. If your Geiger counter is clicking, you're probably taking some amount of radiation damage.

So, this provides you two things:

• General awareness of when radiation damage is being taken.
• An idea of where not to step out of your power armor. (Unless you're wearing a hazmat suit underneath.)

Another thing to be aware of is that damage and max HP reduction from rads is not based on static values in the same way most other damage is. Each rad absorbed takes out 0.1% of your maximum health, regardless of how high or low your natural maximum HP value is.

Rad absorption that's displayed in a numeric form is generally in terms of "rads per second". The greatest available radiation resistance (such as what you can get with power armor) reduces most sources of radiation to "<1" rads per second.

This means you'll usually have somewhere north of 16 minutes and 40 seconds worth of survival time, before you'll have to use Aids or die, while the Geiger counter inside your power armor is clicking away. Certainly, this should be more than enough for most scenarios. But it's still important to be aware that it is a finite period.

One particularly annoying bit about the power suit's radiation gauge however is that it doesn't actually show you how much damage you're taking. Usually, it's extremely low ("<1 rads") but it would be nice to actually see that value so you can be certain.

The suit's HUD also doesn't give you a clear indicator of how much of your maximum health is currently unavailable due to radiation damage already taken. (Well, apparently there is an indicator but it's really easy to miss.) This can be particularly annoying when you are trying to judge how to effectively use your Aids to restore available HP or remove absorbed radiation.

To work around these limitations, you can go into VATS at any time by holding the normal VATS button for a half-second or so. (This works in and out of combat, regardless of whether there's anything around that can be targeted.) There, you'll see the normal health bar and radiation indicators as appropriate.