you can use a splitter so long as the total power draw of all the fans does not equal more that what your mobo puts out, normally that 25w but check your manual, however I would only use a splitter to 2 fans doing the same thing, i.e. 2 case or 2 cpu fans not 1 of each, mainly because the fans will be needed at different speeds at different times.
you can get fans that go straight into the psu but yes, they only run at full tilt and the fans of that style I've used are terrible.
What I would do is find out how many fans your case can hold, and buy a fan controller, there are loads of amazon, then you can as many fans as you like and tell them to shut up if you are just streaming something online