For the most part, an Arduino version and a USB powered version would be the same. The Arduino has a 5v regulated output.
The main problem is, if the schematic you post is accurate, is that 36Ω resistor, at 4.5v, with average led voltages (red 1.8v, yellow 2v, blue 3.3v) is pushing them past the ""standard"" max current draw of 20mA each. Is that 36Ω a typo?
With 5v, its almost double that. The schematic implies with all 10 led/resistor series on, it's taking 151mA, so 15mA each. The math doesn't work.
Current = (Voltage Source - Forward Voltage Drop) / Resistance
Red: (4.5v - (1.8v * 2)) / 36 = 0.025A or 25mA. (5v - (1.8v * 2)) / 36 = 38mA
Yellow: (4.5v - (2v * 2)) / 36 = 13.8mA. (5v - 4v) / 36 = 27mA
Blue: (4.5v - 3.3v) / 36 = 33.3mA. (5v - 3.3v) / 36 = 47.2mA!
Only the yellow would be mostly safe, without increasing the resistance. Simply an extra resistor between 24~68Ω in series with the existing 36Ω for each led strand, would solve this with a maximum draw of 28mA to 16mA per strand. That's how you can protect the leds when using the USB power supply.
Now if you are going to power the leds from the Arduino, directly tied to the Microcontroller's pins, you will quickly eat through the 40mA per pin limit, or 200mA total limit. Especially on the blue section, where you have three leds in parallel. That's 47mA with the existing resistor, less if you add another resistor per led, but you would only be able to have 4 of the sections on at a time. In this case, if you need more, you would need a transistor with a resistor as well.
simulate this circuit – Schematic created using CircuitLab
The SW1 is the manual switch you have. The two 36Ω resistors are the same as a single 72Ω. The three leds are standard 3.3V Blue leds. The NPN transistor allows you to switch more current per pin than directly connecting them would (The 2n3904 tends to allow up to 200mA each). The 1k resistor provides ~4.5mA of current at the transistor base, which should saturate it.
In any case, with a bit of math as listed above, you should have all you need to figure any combination you need out.
Are the batteries protected? Are their overcharge/low voltage cutoffs implemented in circuitry inside the battery which would mean they can be safely used outside of an iPad?
Apple doesn't share technical specs or datasheets on their devices. You are in a better place than most since you can get them cheap and easily. Unless you take one apart, or your suppliers share some data on them, noone knows. Also, beware of those cheap chinese knockoffs. Who knows what shortcuts are taken on them. Apple prices means Apple quality. 10 dollar prices mean 2 dollar quality. You are probably not getting legit 11000mAh batteries at that price point.
Is the fact that the iPad 3/4 battery contains multiple cells going to cause problems with load balancing during use and charging? If this is a problem, the iPad mini contains just one cell and so may be a viable alternative.
Assuming they follow Apple specs, Apple undoubtedly takes load balancing into account. But a single cell is easier since it doesn't require that.
Yes, USB charger is a regulated power supply. USB devices expect around 5 V on the Vcc line. Do note that the regulation on some of the cheaper USB power supplies is pretty bad, especially if you're near the power supply's maximum current.
The best way to be sure if you need a regulator is to check the regulator and what your power supply is providing. If you have a multimeter, use it in the voltage mode on the two outer pins if the USB connector of the power supply. You should get around 5 V.
In the guide you linked to, the regulator used is of 7805 pattern. Here 78 is the series number and 05 is the output voltage of the particular regulator.
Such regulators need higher input voltage of around 7 to 8 volts (2 to 3 volts above the output voltage) to be able to regulate properly. Basically, this type of regulator, called linear regulator, turns the excess voltage into heat and provides 5 V output (for 7805). If the input voltage is too low, it won't be able to give correct output voltage.