Are op-amp circuits more noisy than amplifiers built from discrete components and why?
Why are Op Amps more noisy than other types of amplifiers (BJT and MOSFET, for example)?


I doubt this is true, especially if you consider the noise factor for the same circuit gain (but another answer might give more concrete information). 


  So, in short, why do OP Amps have electrical noise issues (if at all), and what is the source?


TI has a couple of good app notes on this: 

Chapter 10 of Op-amps for Everyone

Noise Analysis for High-speed Op-amps

In short, the feedback resistors and the op-amp's input stage transistors both contribute to noise. 


  Perhaps Op Amps lack sufficient filtering capabilities that are inherent to BJT and MOSFET design?


Op-amp circuits can be filtered to whatever bandwidth you like. It only requires adding an extra capacitor or two to the op-amp circuit, or for higher-order filtering adding an additional filter stage after the high-gain stage. Which would be equally required for a discrete circuit.