The passenger side valve cover on my Vette with new 383 stroked engine doesn't have a breather hose going from the hole in the passenger side valve cover up to the bottom of the air filter. Could the lack of this hose cause my idle to be slightly erratic? What about the opening in the valve cover? Couldn't dirt be getting sucked in there? Should I plug it up?

DO NOT PLUG IT UP! Sight unseen, it's difficult for me to really say for sure, but I think you will want to install a breather cap on that opening, or a grommet and adapter to connect the hose to the air cleaner.

Correct, that hose/hole is the intake for fresh air for your PCV system. If you plug it, you will cause all the blowby gasses in the engine to build up, due to lack of air flow. This will also leave condensation that builds in the engine. The result is a corrosive acid substance that will attact the internal components of the engine.
All you need is a filter. You can get one at many parts stores or speed shops. It lookes like a tall oil filler cap and has a filter material inside. Or you could hook up the hose.
This will not affect the idle of the car, unless you plug it. The idle problem is somewhere else. If all the cylinders are firing, check for vacuum leaks or a misadjusted carb.

Hmmm...well the guy at Kragen's told me to install a Billet cap. Looks like a chrome plated rubber plug. He said the PCV Valve is on the driver's side valve cover and that as long as I have a PCV valve there with a hose to the carb, I'm fine. He told me to plug up the hole on the passenger side valve cover the same as if a oil filler cap was on it. Actually, isn't that where the oil should go? If not, where do I pour oil into the motor?

You do need an oil filler. Those do close. But you still need an air intake for the PCV. Without it, you will have problems. If you don't see problems it will still affect oil and engine life. If you only have one opening in the right valve cover, you can use a vented oil filler cap.