As you say, that Cooler Master case comes with its own ARGB Controller board you mount inside the case. To that you connect a power supply from a PSU SATA power output, a communication cable from a Controller socket to a mobo USB2 header, a cable from the board to the case's control button mounted at top front, and a signal cable from a board connector to one of your mobo's 3-pin ARGB headers. Apparently you can have control of the lights by this board done either by using the USB2 communication cable and Cooler Master software you download and install, OR by the mobo's software using the connection to the mobo ARGB header. What is NOT clear in the manual is how you make this choice. MAYBE you can connect both signal sources and use some setting in the Cooler Master software to set which control signal is actually used. MAYBE such a choice depends on using the case top front switch. Or MAYBE you only get to use ONE signal source, and you must connect only one of those signal cables to the board. You really ought to ask Cooler Master how that is supposed to be done.
Regarding the Thermaltake TT Sync controller box, my reading is that this unit can be connected to any standard 3-pin ARGB signal source. What it does is use its own power supply from a PSU SATA output so it does not actually draw power for lights from the mobo header (or whichever source) - it only uses those signals as information to direct its own outputs. NOTE that it specifies that, since the SATA power source is limited to 5 A max, that is the limit for the total load on ALL of the output ports of this Controller box. So IF you wish to have the mobo's software tools do control of all your ARGB lighting devices, you can connect a cable from its 3-pin ARGB header to the input of the TT Sync box and plug all your devices into that box. Now that brings up another question. The TT Sync box has several 9-pin output sockets, each able to supply up to three ARGB devices. For this they supply cables from the TT Sync unit to ARGB devices. But the connectors to those ARGB devices use 4 pins, so I am not sure whether these are completely compatible with other makers' common 3-pin ARGB lighting devices. You need to ask Thermaltake for that detail.
IF you were to consider using both the Cooler Master case's Controller board for its case fan ARGB components and the TT Sync box for other ARGB lighting units, BOTH controlled by the output signals from the mobo 3-pin ARGB header, then yes, the simplest way is to use a RGB Splitter cable to feed that same mobo signal to both Controller boxes. Now, the common RGB splitter cables are designed for 4-pin plain RGB systems, but they also fit 3-pin ARGB systems I believe because the pin size and spacing is the same. You just end up not using one of the connections in such a cable. BUT two things re VITAL. First, you must never make the mistake of plugging such a cable into a mobo 4-pin plain RGB header because that would be possible but damaging. And secondly, when you use it, you MUST make sure you turn the connectors the right way so you get the connections at every junction correct so that the +5 VDC line is always matched up, etc.
Regarding your question #2, you are right. The ARGB Splitter cable supplied with the Cooler Master case and Controller board is simply a way to connect two ARGB devices (in the front fans) to a single standard ARGB port. That does not have to be a port on that Controller card, although the connector on the Controller end of that Splittter cable may not match up with some other maker's connector. But if your intention is to have all of your ARGB devices controlled by the same signals from the mobo ARGB header, you do not have to change that. The fact that the same mobo signals are being fed to two different Controller boards means that both boards will do the same thing.