Are some bits only for plow reining and some bits for neck reining? All this rain has me sitting inside thinking...

I think you can do either based on how well the horse is trained.  Typically a plow rein type bit is one like a ring snaffle where when you pull on one side of the bit, it takes the other side with it, pulling the horses head that way.  A neck rein but is usually one with shanks that when you lay the reins on the neck, there isn't much going on with the bit.    You can lay a rein on the neck in a snaffle, halter, or hackamore and get the same effect on a neck rein trained horse. You can also pretty well plow rein a horse the same way, but depending on the bit's make-up, you might not get a lot of response. 

Barrel racing bits are typically designed to flex so that you get one side of the bit doing one thing (IE: Lifting) and the other side is neutral.   

Reining or roping bits are made to stop or use neck reining pressure to get the horse to change direction.  

Yes. Some bits are going to have a nice clear signal with a direct rein. Others are not intended for a direct rein, so if you try to use it in that way, you'll send conflicting signals. Can a well-trained horse handle that? Usually.

You have to think about what is going to happen on the off-side of the bit if you pick up a direct rein. Is it going to be cueing the horse in some way that is conflicting with the direct rein you are asking for?

Wanting names of mouthpieces/shanks that work one handed as opposed to those that need direct reins. 

A curb-bit with a shank is typically a one handed bit.   They can have various mouthpieces as long as they don't have much gag action.

Gag Action creates some issues with one handed bits.  When you lay against the neck the pull can go lots of different ways.  See below