BIOS and software measure at 2 different points on the CPU.
If you are reading the package temp on a program then it will be much lower that core temp.
Your temps are mostly fine, when you start exceeding 65C on either.
So I am good? I don't need to be freaking out and buying a 3rd party cooler or a new motherboard?
Nope, you don't NEED to... But a cheapish (30$) aftermarket cooler is NEVER a bad idea on an AMD setup.
A Xingmatic Gaia or Coolermaster Hyper 212 EVO are great options for very little money.