The discharge rate could be an issue, so that's one reason going with 15Ah is probably better. However, the higher capacity battery could be an issue for the charger, depending on how that works. But 14 is pretty close to 15, so I would just go with the 15Ah. I would be very surprised if the 12Ah didn't work, though, so if that somehow fits better or is much cheaper, it should be fine (don't forget the 15 will give you 25% more run time than the 12).
Most NiMH chemistries will probably tolerate at least a few C discharge rate, so an AA should be able to deliver at least 5A, which is plenty for the motor you're describing.
I have no idea what you're asking with this solar stuff, probably because you have little idea. Among your likely confusions: weight is not at all enough to tell us how much power you need, and getting more current available in your batteries is not going to force more current into your motor.
You should not be putting batteries in parallel unless you really know what you are doing, which in this case you do not.
Maximum, optimal, and safe discharge rates are completely separate. Any maximum rating better still be safe, and there will be a wide range of safe discharge rates. Optimal in terms of energy delivered will probably be around a few mA over a month, maximum could be 20A for a few minutes.
Thanks for the feedback. It looks like you are aware that this is not the most appropriate avenue for this discussion, so you might want to think about what prompted you to post here before trying to contact us some more conventional way. You do bring up two subjects that I think are worth discussing publicly.
First and most directly addressing your comment, you got quoted a little over $500 for over 2500 identical little parts. A laser cutter is not necessarily going to be the most cost-effective way to do that kind of job, and if you have a better way of doing it, great: do it that better way. It's a similar situation with the competing quote: I don't think we promise being the lowest-cost option, but you might not be getting an apples-to-apples comparison since there are other factors like turn time, quality, support, etc. I'm not saying the cheaper options are necessarily going to turn out worse: I might have done your job for half or less back when I was running a laser out of my house. Still, a lot of your ire seems to be stemming from your disbelief about it taking four hours, and that does not sound that off to me.
Second and more troubling is your resorting to complaining about greed, which seems all too popular lately. There's some rate at which things are worth it for us to do, and if that rate does not make sense for your application, don't go for it. There are probably things that you would not do for $100 but that you would do for $1,000; does that make you greedy (in some bad way)? You say, "Might as well get them all cut, right?"; isn't that the same greediness? It's fine for you to come back to us with your other quotes to make sure there wasn't some miscommunication about your requirements or to try to negotiate a better price, but it's not appropriate or helpful to stoop to that kind of name-calling. Also, you should want to work and interact with competent people, for whom "greedy" probably isn't a particularly meaningful word.
The Maestros should give very good, smooth outputs, so the jerkiness is probably because of the servo you're using or the load you have on it. You're probably effectively getting a situation like the one in the last two diagrams, where the commanded position is gradually changing but the servo responds in jerks either because it isn't very good or because you have so much load on it that it can't move until the error is quite big. Can you put the same servo setup on an RC receiver and slowly move the controls on your transmitter?
I want my office near Pololu Valley! We have customers on the Big Island, so it might be possible to see some of our products in action there (I remember hearing some talk of a robot club in Waimea a while ago, but I don't know what came of that). I don't think we have any distributors in Hawaii.
The main motivation is cheap, relatively easy control: you can use just a single I/O line on a microcontroller, and you don't have to deal with any power electronics to drive the motor. If you want a motor controller that is more than just an H-bridge that you might have to do a lot of work with, it's difficult to get that and the motor and gearbox for the ~$10 of a cheap servo.
The servo is not made for continuous rotation, and it rotating continuously is a sign that you are asking it to go beyond its limit, which could cause it to be destroyed. The standard 1-2 ms range should get you close to the full turn, and you can gradually expand the range while keeping it centered at 1.5 ms to get to exactly one rotation of range.
By the way, you should direct questions like these to our forums or to more direct tech support.