You're kind of all over the place there, so I will just give you bullet point comments as I go through your post:
* I don't know where you're coming up with this "(as you said)" note.
* All kinds of devices (including motors and light bulbs) draw less current when you give them less voltage. You have to have some somewhat special electronics to draw more current when the voltage goes down. Some LED flashlights could certainly be in the "somewhat special electronics" category.
* I'm used to RC toy cars running for maybe 10 minutes, so 23 extra minutes is huge. Obviously, it's not in your case, but you should see it's useless for you to just mention 23 minutes without some baseline.
* Your Application 1 text is difficult to follow and seems to have a lot of extra unnecessary details.
* If you are asking for confirmation that you cannot calculate the battery life without knowing the current you are drawing, then yeah. But in that case, I don't know where you are getting your confidence about running off a particular battery configuration.
* If 1W is your input power at 12 V, you can figure 1/12 A (or 83 mA) for your current.
* I don't think a lot of little LEDs is an efficient way to get a lot of light. That's based on the good flashlights not seeming to do that.
* I expect a decent AA NiMH cell to be able to put out 7 amps for a few seconds, but if you really care, you should try to get the specs for your particular batteries.
What does your question have to do with continuous-rotation or multi-turn servos (the topic of this post)? Also, it would be nice if you didn't call your servos "servo motors" given that one of the posts in the series details why not to call them servo motors.
Anyway, it's completely possible for you to destroy your servos by giving them a bad signal. You should look at your servo control signal with an oscilloscope to see what your program is doing.
Did you read the post? Anyway, 2000 mAh should last about 1 hour at a 2 A discharge rate, 2 hours at a 1 A discharge rate, 10 hours at a 200 mA discharge rate, and so on. Typically, the capacity is based on some particular discharge rate, and capacity will be lower if you discharge faster than that. By the way, a modern cell phone battery is going to be a lithium-based battery, which you have to be extra careful not to over-discharge, and the battery module might have some integrated electronics to protect and monitor the battery.
Thank you very much for your feedback. At this point, I think that if I could just flip a switch and have comments in a way that would not be that objectionable (for instance, with some of the suggestions you had), I would turn that feature on. The way we're organized right now, though, our few web programmers are also involved in product design, and I would rather spend our time on new products or on more important features for the web site (such as the search system).
In general, it's pretty difficult to change a normal servo to a multi-turn servo; at the very least, you need to change your feedback potentiometer to a multi-turn pot or change the gearing to it somehow.
I do not know what you mean by the "in one direction" part. If it can go more than 360 degrees and keep track of position, it will be able to do that in both directions.
This article is about using one PWM source to get multiple servo control signals. As I said in the post, I think it is not helpful to think of servo control pulses as general PWM signals. You cannot get four arbitrary PWM signals out of one PWM signal.
Your basic math is right, but keep in mind that capacity depends on discharge rate, so depending on the battery and how it's rated, it might not necessarily give you the 4320 mAh if you discharge it at 4.3 A. We do not carry any packs with that high of a capacity, but that's in the range of what you would get with "C" sized NiMH cells. You should be able to find receiver or servo packs with that kind of capacity at a hobby store (a quick search at Tower Hobbies led to http://www3.towerhobbies.com/cgi-bin/wti0001p?&I=LXVTG1&P=7 ; at first glance, it might be what you're looking for). You could also consider lead-acid batteries; they will be bulkier, but they will probably be cheaper and easier to find in capacities exceeding 4Ah.
I am not suggesting that "qualified and accountable" people should not be involved. However, I think that government should not have a monopoly on safety standards and that the existing government monopoly leads to sub-optimal results (just look at the post above yours for an example). For instance, we could have facilities rated by some private company, and employees could choose whether to work in a place that was not rated or had a lower rating than they were comfortable with. Your insurance example shows yet another way the private market incentivizes safety. (And doesn't your suggestion of going to the insurance company indicate that you think the private company might have been more effective than the government?)
Thanks for that perspective. I am generally in favor of leaving room for discretion, but that leads to more room for abuse because of the government's monopoly. If we instead had private inspections (which the county basically deferred to, anyway), there would be a little more room for alternatives if one particular inspector were being unreasonable.