When it is set to the prime position, it's just an open valve, flowing fuel freely to the carbs. So it will work with the line off. When the line is hooked up, it pulls a vacuum, which pulls on a diaphragm in the petcock opening the valve when set to 'on' or 'reserve'.
There could be a number of things causing the crazyness with the high idle with the enrichment circuit on. Even though it works fine in the normal position, you should still sort out that run away rev issue, because it's not supposed to do that.
Basically the choke is only need on a cold motor. You can look up the specifics if you care too on google, but in short the cylinder walls are cold so you have issues with fuel and condensation. The choke will add a bit more fuel to compensate. It will run fine, then as it starts to warm up after a minute it will start to want to bog/die since its getting too rich. When you turn off the choke at that point, the mixture will go leaner to a proper air/fuel ratio.
What could be happening with your bike might be related to the line being off, might have one carb with a clogged up enrichment circuit, might just be out of sync with each other real bad, carbs could be all dirty/gunked up inside.
Bottom line is, when you get a new bike the first thing you want to do, and I think everyone that's owned more than one machine will agree, is to pull the carbs off no matter what. Take them apart, toss a rebuild kit on them, give all the internals a good look, clean them a few times, even if you think they are clean they are not, check float heights, check that the jets are correct sizes, check the slide diaphragms for holes/cracks, check the gaskets and o rings and seals(looks like you have a leaky shaft seal from the pictures you posted), replace fuel lines, add a fuel filter, make sure airfilters are clean, check carb boots for cracks, check plugs and plug wires.
That's some general fuel/air system basics that I do every time I buy a carbed vehicle. You'd be surprised what you will find, no matter what the original owner has said they have done.
So pull them carbs, crack open a beer, and get learning! We can walk you through any part of the process.