You need 1 for every fan you want to plug into the motherboard. The fans that come in many cases are powered right of the psu so they don't use any motherboard connectors. On my system I only use 2 so I can control the 2 fans on my H80 cpu cooler, the rest of the case fans are power by molex connectors off the powersupply.
The only fan connector that is essential on a motherboard is the cpu fan connector.
Every motherboard has one. You are good.
It is possible to connect all fans to the psu.
What a motherboard header gives you is the ability to vary the fan speeds.
As to how many fans are needed, really, not many.
If you have two 120mm intake fans, or the equivalent in output fans, then you will have adequate airflow.
What you really are trying to do is get the hot air out of the case.
If you have a decent cooling case, I would not get an all-in-one liquid cooler.
They are expensive, noisy, less reliable, and do not cool any better than a top cooler.
Why spend $100 on a cooler so you can oc 0.1ghz higher, when that money would be better spent on a faster cpu or graphics card in the first place. For most, a $30 cm hyper212 will be very good.