Can this motherboard support this cooler?

The B350M Bazooka does not have a "pump" header, but, these headers are technically similar to the typical fan headers where you can also plug your AIO into. Some motherboards just label these headers as "pump" (but still can be used with a fan), while some motherboards designate these "pump" headers due to the header having a much higher current output (usually 2A and above) compared to normal fan headers (usually 1A).

So, you can safely plug in the CM MasterLiquid Lite pump into the nearest fan header of the B350 Bazooka (say, the SYS_FAN1 or the SYS_FAN2). Note that the CM pump has a 3-pin (DC) connector. The B350 Bazooka, by default, has the SYS_FAN1 and SYS_FAN2 running in DC-mode. You have the option to change the mode in BIOS. Using a 3-pin (DC) fan or pump on a mobo header in DC-mode would mean that the speed of the fan or pump can be changed. Using a 3-pin (DC) fan or pump on a mobo header in PWM-mode would mean that the speed of the fan or pump will always run full speed (due to constant 12V power being supplied). Most recommend that pumps, especially non-variable pumps such as in mainstream AIOs, be run in constant speed -- as varying the pump cycle/rate may lead to an increase wear-and-tear of the pump's life.

As for the radiator fan, it is advisable to plug it in the CPU_FAN header, which is by defualt, in PWM-mode. The CM fan included in your AIO is a 4-pin (PWM) fan, and so, it should be run in that PWM header. This way, the fan will only ramp up its speed when needed.

Since your friend wants to use two (2) additional case fans, and he is only left with one (1) available header, then, yes, he should get a splitter. However, it is important to take note of the current rating of the fans he wants to connect (the amperes each fan is rated to draw can be seen in its box or sticker behind the fan). Ensure that the total current of the two fans will not exceed 75% of the single fan header rating (so, ~0.75A total or less for both case fans). Using fans that total to much more of the ampere rating may damage the motherboard (or portions of it).