As a rough approximation (within a factor of two), I would expect similar battery life. The main thing I'm skeptical about is that 5V is not a natural voltage for a lithium ion battery. If there's actually a 3.7 V, 2600 mAh battery inside and the converter to 5V is 90% efficient, you'd be getting the equivalent of about 5V, 1750mAh.
I don't know enough about your system to know for sure. There's got to be some limit on your alternator and whatever battery charging circuit is in your car, and while my impression is that lead acid batteries are among the more forgiving regarding charging, you still might limit their life or otherwise damage them if you just do what you are proposing.
I am concerned that with the level of electronics understanding you have presented, you might be risking damaging or destroying some fairly expensive equipment. It seems like you might be better off looking for some pre-made solution, like a general-purpose external/backup battery for phones or other electronics.
On to your specific case. First off, you should realize that with four batteries instead of the built-in two, you will get at most 2x the battery life. You should not put your batteries in parallel, and if you just put all your batteries in parallel, you would only have the 1.2V individual battery voltage anyway. You should also go through a regulator, too, to make sure you actually give your device the 5V it is expecting since the fully charged batteries might be 5.5V or more and damage the device, and they might get too low as they discharge. Something like this step-up/step-down regulator could work:
With that regulator, it would be better to go with 6 AA batteries if you have the room for it. But, if you go down this path, make sure you get a better understanding of what you are doing and that you are ready for the consequences if you make a mistake.
I don't know what that 12A/5h notation indicates. My first guess is that it's a typo or other mistake; my second one is that it's 12Ah at a five hour discharge rate. If the batteries weigh about the same and have similar dimensions, their capacities will be similar.
"How many mAh a battery will last" is not a good question formulation because it sounds like you want to know how long a battery will last just based on its capacity. It's like asking how many days until you need to refuel your car that has a 12-gallon gas tank without saying anything about how much you drive per day.
As I said in the 9V alkaline battery example, you're basically going to get about the same amount of energy per weight for any given battery chemistry, so since a 9V weighs about the same as a AA, it's going to have about the same amount of energy, and since energy is measured in Wh, which is Ah times V, a 9V battery will have about six times lower mAh rating than a 1.5V battery of the same weight. If you happen to know your application for the 9V draws the same current as a specific application for AA or AAA, then yeah, it will not last as long, but that kind of situation is rare since there aren't many products that can run from either 9V or 1.5V.
I am happy to hear you have had good experiences with us. Lynxmotion has sentimental value for me since I played with some of their earliest arms and hexapods almost 20 years ago, but I have not really kept up with their more recent offerings. Is there something you see RobotShop actively doing to hurt Lynxmotion? My impression was that Lynxmotion was mostly the effort of its founder and that he retired; if that's the case, RobotShop would be saving it from disappearing completely rather than causing a problem. But I have no inside information and have not even paid attention to public information about it, so for all I know, all the original Lynxmotion folks could still be active in the company and butting heads with their new parent company.
I would like to offer more of the servo-based parts and kits you are asking about, but there is nothing specific in the works, so it's unlikely that we would have anything new for at least the next few months. We might start with something basic like small grippers; for bigger, more complicated systems, it's not clear to me that general-purpose RC servos are the way to go. Something like the Robotis/Bioloid products, with specialized (and proprietary) servos offer a lot more for robotics. How do you think these newer systems compare to the Lynxmotion type of products?
Your question has nothing to do with servos, but briefly: cheap RC toys often have separate frequencies, like 27MHz and 49MHz, to avoid interference. More sophisticated RC systems use several dozen channels in the 72MHz and 75MHz frequencies, and you have to use different channels if you want to prevent interference between multiple systems operating at the same time and place. There are also rules about which frequencies can be used for surface models and which can be used for aircraft; these things can all vary by country. Newer 2.4GHz spread spectrum systems automatically jump around several channels, so multiple systems are not supposed to interfere with each other (you will have to do some kind of "binding" process so that a particular receiver knows which particular transmitter to listen to).
The 9.81 in your calculations indicates you are getting thrown off by thinking about the force being in Newtons, but I think you are better off thinking of your 2kg as a force since stepper motor torques are commonly specified in kg-cm, anyway. So, if you have a 2.5cm radius pulley wheel directly on your motor and hang 2kg on it, you would need 5 kg-cm to hold it. Keep in mind that the holding torque that is usually specified is a stepper motor's strongest torque, so you will need a motor with a much higher holding torque if you want to be able to lift the 2kg instead of just hold it in place.
The 0.5 factor you have in one of your alternatives makes me nervous that you are talking about some complicated system of pulleys instead of just the single wheel on the motor shaft (what I called a spool in the article). If that is your situation, you would have to say what that pulley arrangement is for anyone to be able to calculate the forces and torques involved.