We'll deal in order.
1. If you connect that fan to a lower voltage source, it WILL work but at lower speed IF that voltage is high enough. For your 12V fan, a 9V supply is certainly enough and that will work. (It might drain the 9V battery, though, because those little batteries are made for small loads.) The 5 V battery might not work, though. Every fan motor has a MINIMUM voltage needed to actually start it up from stalled, and it happens that many computer 12 V fans cannot be started with as little as 5 V.
2. If you have three mobo SYS_FAN or CHA_FAN headers to use for your three fans, then certainly go ahead and simply use one fan per header. Just remember that there are two rather different groups of mobo headers and fan uses. ONE fan (sometimes more) is used to cool the CPU chip, and it always should be connected to the CPU_FAN header. That header is always guided in its automatic speed control by a temperature sensor inside the CPU chip itself. In addition, that header gives extra attention to monitoring its fan for possible failure and will take quick action if it suddenly finds its fan is not working. All of the CASE ventilation fans, however, should be connected to SYS_FAN or CHA_FAN headers. These normally are guided by a different temp sensor on the mobo, BUT some mobos give you the option to use another sensor, and a few even arrive set to use the internal CPU temp sensor, which is wrong.
3. If you opt to use a Splitter to connect two or more fans to a single header, it should not involve a Molex connector like the ones on that three-headed item in your photo. A fan Splitter has two (or more) male output "arms" (with pins) for your fans, and only one "arm" ending in a very similar female fan connector with three holes that plugs into the mobo header. A Splitter is a simple device that just connects all its fans in parallel to the header, and draws all the power for its fans from that header. The header has a limit (usually 1.0 A) for the max current it can supply to ALL fans connected to it, so this limit is a factor when using a Splitter. By far MOST modern computer fans use at max 0.10 to 0.25 A at max, so three fans is almost always OK. HOWEVER, there is a lot of confusion because there is another type of device called a HUB that can be used to connect several fans to one header. Unfortunately, sellers tend to misuse the labels Splitter and Hub as if they were interchangeable, and they actually are different. A HUB has one extra "arm" that must plug into a power output from the PSU (either a 4-pin Molex or a SATA connector), and the Hub gets all power for its fans from the PSU, thus avoiding the current limit of the host fan header. BUT that type of device can NOT control the speed of 3-pin fans, which is what you have. So do NOT get a fan Hub; stick with a Splitter if you connect more than one fan to a header. You may find a 3-pin Splitter hard to find, but any 4-pin Splitter also will work just as well.
4. You have spotted an important thing to know. Any mobo fan header processes the speed signal sent back to it (it is a series of pulses - 2 per revolution - sent to the header) by counting the pulses. As you anticipate, that only works if the header receives just ONE train of pulses - it cannot deal with two or more speed pulse trains from several fans. So any Splitter or Hub will send back to its host header only ONE fan's speed signal. Since the signal is sent on Pin #3, the simple means for this is that, among the output male connectors of the Splitter, only one has a Pin #3 to make the connection, and the other output(s) are missing that pin. Those "others" look exactly like the one in the middle of your photo. (The one in the photo is missing that pin for a different reason - the device is to connect for power from the PSU, and there is NO reason to send a fan speed signal back to the PSU that has NO way to accept and process it.)
I'll add a further note, since your fans are of the 3-pin design. That is the older type in which the fan motor speed is controlled by varying the VOLTAGE supplied to it on Pin #2. The newer fan design now is called PWM, and uses a very similar connector with a fourth pin or hole. The two types have been designed for some ability to tolerate mis-matching, with the result that you can plug either fan type into any 3- or 4-pin mobo header and it will work - sort of. The one mis-match that needs a bit of attention affects you because you have 3-pin fans. If you plug that into a 4-pin header that is using the new PWM Mode for controlling its fan, your 3-pin fan will always run full speed. BUT almost all mobos now use 4-pin headers AND allow you, in the BIOS Setup screens for configuring the individual fan headers, an option to set the header to use Voltage Control Mode (aka DC Mode) instead of the new PWM Mode. This WILL control the speed of your 3-pin fan. So, when you get things assembled, you can look for that BIOS Setup option and make the adjustment.