Modern PSU fans only spin when on load. Other times, PSU fan not spinning could be due to low load, connection issues, dust build-up, or a malfunction.
PSU
You can easily use online power supply calculator to know how much power your PC needs. Another way is to check GPU manufacturer's website...
A fanless PSU is only worthy if you want to reduce the overall noise coming from your PC. Here're the drawbacks of going fanless.. Read more..
No, you can’t mix PSU cables. The difference in pinout can blow your PC. However, you can still mix them if it's in the compatibility list.
Learn why you should not neglect your power supply and settle for a cheap one. Low grade PSUs are more prone to failure and can damage your entire PC.