A regulator drops the propane from the high-pressure in the tank to the low-pressure used by appliances. Your system will only work if you have one two-stage regulator in the system between the propane tank and the appliance. Two regulators (one at the tank and the other at the appliance) won't work.
Typically, the regulator is at or near the tank, and you'd run a low-pressure line (3/8" to 3/4") from the regulator to each appliance using either a manifold or several Ts. Just make sure your regulator is appropriately sized, a tiny regulator will "freeze up" if you try to do too much with it.
However, it is acceptable to run a high pressure line (1/4" to 3/8") direct to each appliance and put a regulator at the appliance.
Regardless, you need to do this right. If you create leaks in the system, you could injure or kill yourself.
Okay, so I took some pictures and have started looking into this more. I pulled everything from the same RV and am trying to to replicate the setup, as they were running the stove, water heater, and an air heater on this setup (EDIT: And a propane fridge, forgot about that). I have a different HVAC heating system, so just the stove and water heater need to run from this.
Here is the tank, with the regulator:
And I had pulled off the original piping, because I intended to match it up. You can see here the original piping next to the 3/8" piping that I THOUGHT was the same:
Here is the water heater (I attached the 1/4" line to play around with it, the line isn't attached on the other side):
And finally, the LP inlet at the stove:
Putting this all together helps me greatly. It looks like I was incorrect about the original main line sizing. It's actually 1/2" (looks like), but it has a 1/2" flare fitting (fits with 3/8" pipe, to intentionally confuse me) on the tank-side that I was going off of, so it steps up from 1/2" flare to 5/8" flare, most likely, and onto a 1/2" pipe. That pipe then feeds everything.
Now, about the water heater. The inlet is threaded to 3/8" fitting, seen in the picture, which goes with 1/4" pipe. So if the water heater truly need 3/8" nominal pipe, I can run 3/8" to it and will have to put in an adapter from 1/2" to 3/8". Not impossible, but I dislike having many connections (read: leak points).
And the stove, which according to here (PDF warning):
https://www.suppliesdepot.com/images/product-detail/MAXICOE00005_1-4.pdf
has a 1/4" inlet and a maximum pressure of 1/2 PSI.
So, all in all, looks like I have to rethink this. I know I'm not a professional at LP and I'm taking a risk, but part of the fun of the conversion stuff is the learning, and if I have someone do it for me, I'll never learn. That being said, if I can't figure this out, I'll swing back to the junkyard I pulled these parts from and take a closer look at the connections left behind (if they're still there) or call a professional.