Does Night Shift mode improve battery life?

This depends on how Night Shift works "under the hood". If post render color shifting is all it does, then it's impact on battery is likely negligible. However I'd expect it to underperform more if that were the case. A filter by itself would likely feel less subtle, throughout your entire experience, than Night Shift does. I suspect additional logic, throughout the rendering process, is used. Depending on how that impacts what is normally abstracted away for optimization, it would be easy for this to have an impact on battery life. Since Apple tried to prevent its use while in low power mode, I am even more inclined to believe this is the case.

It does neither. Changing the screen tone to a more yellowish / warmer hue will have have no effect on battery life either way. Night Shift is merely for your perception. It has no effect on the device itself.

Actually, with screen being the most battery-hungry peripheral, and all battery test being conducted under a very specific brightness settings, it stands to reason that applying any filter to the whole picture will affect the battery usage.

It also makes power saving auto-disabling the night shift seem less like a bug.

I sort of agree but have no proof.

Yep, me neither. Battery seems "weaker" since 9.3 update, not sure why. Night Shift might be an explanation, as I'm using it heavily.

No. It can drain your battery if you use it with "sunset-sunrise" scheduled option, nothing rilevant (maybe 1% drains over all day?), but you'll have another background service who's running in order to check your location and sunet time. Manual setting doesn't affect your battery.