Why is a solid state drive needed as well as a hard drive?
It's not needed per se, but it's a really nice thing to have. An SSD uses different tech compared to a hard drive and is much faster. If you install the OS one one the computer will feel much faster than when using just a hard drive. It's definitely worth the extra money.
Does the CPU fan come with the motherboard?
No, but it comes with the CPU. This is because different processors output different amounts of heat and thus need more or less cooling.
How are USB ports that come with the case connected to the motherboard, is there a connector?
Yes, there are two cables inside the case, one for the two USB2 ports, and one for the single USB3 port. They plug into the marked connectors on the motherboard.
The GeForce GTX 960 video card requires 350W PSU, is that extra to the man 450W PSU? Also, isn't the video card powered by the motherboard?
No, that's for the entire system. The 450W XFX PSU I picked out is actually a bit overkill, but it was one of the cheapest good power supplies I could find when I put your build together.
Can I use any power supply for any case (wattage wise)?
As long as both use the same mounting holes and physically fits, yes.
Does anybody have any experience with the Corsair vs550 PSU? Is it good?
The Corsair VS is cheaper than the Corsair CX, which is already a budget PSU, so I'd stay away from that one if I were you. It's not like the XFX I showed you is expensive or anything, £34 is pretty damn cheap for a quality power supply.