"I know from my shot shell reloading that I can't swap out different hulls and wads without changing the pressure of the load. Is that a concern with pistol and rifle as well? "
Yes
"If I get a shipment of mixed brass do I need to sort it and develop different loads for the different brass?"
For 5.56 mil spec rifle brass, there is probably value in sorting cases, but if they are same headstamp from different lots and years, consistency may still vary, perhaps just less so. If you are using mixed brass I would avoid max loads. If they are all truly once fired, and not machine gun fired, the main thing is consistency in processing and you should be good for plinking. Inspect the brass for neck splits and other problems as you use them. Get a good quality sizing die and a case gage (Dillon or similar) to as a check on specifications.
"Also, do I need a load that is specific to my particular bullet or will all FMJ bullets of the same weight be interchangeable. Thanks for you help in keeping all my fingers in the right place."
Bullet style and quality is in my experience the number one factor in accuracy. For .223 55 gr and 62 gr mil spec bullets will be similar, there will still be differences. If you're not running max loads, you should be able to use mil spec bullets of same weight almost interchangeably, but accuracy will vary. However for safety and sound practice, you should work up a load for a bullet then use caution when you change to another maker.
For handgun, things are different for semi auto vs revolver cartridges. If you need to crimp in a bullet groove or cannelure for safe revolver shooting, then consistent case length is important and you might have to trim with brass by different makers, maybe at least once.
Shooting range or once fired mixed brass in 9mm or .45 ACP and similar designs is pretty normal, but best if you know the source of the brass, and that they are truly once fired if advertised as such. A good taper crimp is what you need for them, best if done in separate final or 4th step. Trimming is almost never needed for 9mm, .45 ACP or similar design. Loads for same bullet style and wt can be virtually interchangeable, but again there will still be differences, and if up near a max load, use caution and precautions.