Yes, that Splitter will do well. If you look at upper left of that page, there are five photos to choose. The last (bottom) one shows a 4-pin fan plugged into one output. Compare that to the photo above it with a 3-pin fan plugged in. Note that the only difference is that the connector of the 3-pin fan is not so wide and does not use Pin #4 of the output. That pin is for the PWM signal that a 3-pin fan cannot use, so it is simply not part of the connection for that fan style.
Regarding fan speed, BOTH 3-pin and 4-pin fans CAN be controlled by mobo headers, BUT to do that the header MUST use the correct control mechanism or MODE. Your mobo has all the settings you need in BIOS Setup to accomplish this. See its manual, p. 26 under "Smart Fan 5 Settings". For "Monitor", select the particular SYS_FAN header you want to adjust - you'll need to make these configuration settings for EACH header you use for the 3-pin case fans. Now for each, set it to use the "Normal" Fan Speed Control (so it does its work automatically), and the common Motherboard temperature sensor so the fans are guided by that sensor. (A mobo has a temp sensor inside the CPU chip to guide the CPU_FAN header, and one or more sensors on the mobo to guide the SYS_FAN headers. There are a few that sense temperature of specific mobo components like the voltage regulator or North Bridge chips IF you feel a special need to use them, but the general Motherboard temp sensor is best for case ventilation.) Then under "Fan Control Mode" choose "Voltage", which is required for 3-pin fans. When done with all the SYS_FAN headers, remember to SAVE and EXIT so your settings are saved and the system reboots.
The Corsair Commander Pro is their most advanced hub for control of both fan speeds and LED lighting displays using their software tool, iCue. It does give you a LOT of lighting display possibilities, and claims to be able to control the speeds of both 3- and 4-pin fans. Regarding fan speed control only, it does not give you anything more than you can achieve using the mobo configuration options I've detailed above.
You have not given us info on how you plan to control the RGB lighting effects of the SP120 RGB fans - 3 with case, plus 3 added, right? Maybe the case came also with the Corsair RGB controller Hub and Controller modules. This was an earlier RGB display control system, entirely manual. It consists of a Hub with six output connectors to the RGB lights on fans, plus a connection the a PSU output for power and a socket to plug in the separate "controller" box. It also has a couple of sockets for optional connection to case buttons. It has NO way to accept external control signals from a mobo header or software tool - your only method of controlling the RGB lighting effects is to use the manual "controller" box and its 3 buttons. Other Corsair products (the Commander Pro being their latest) can do these things and much more using software tools. But with this included control system, you plug each fan's RGB connector cable into the Hub unit (note the need for a continuous sequence of ports in use) and then the "Controller" box's buttons allow you to choose manually from several items for speed, colour and mode of the RGB displays. ALL of the fans' lighting modes will be controlled in the same way - there is only one output "channel" - but some displays involve progressive changes among the fans. If that gives you all the control you want for the RGB lighting effects, then you do not need 'the Commander Pro for that, and you don't need it for fan speed control, either. In fact, I am not clear on whether the Commander Pro even can control the RGB systems in the SP120 RGB fans. Their wesbsite for the Commander Pro appears to say yes, but the RGB type does not match. The Commander Pro has two output ports for what are called ADDR RGB LED systems, whereas the lighting systems in the SP120 RGB fans appears to be of the other simpler RGB 5050 systems, and the RGB Hub included with those fans does not appear to be able to accept control signals from an external source. The Corsair website seems to suggest that this can be done if you buy also their middle-level RGB Controller, the Lighting Node Pro, and use that instead of the base-level Hub and Controller system you have already, but I am not sure about that.