You have some newbie issues. First, you'll want to understand that each 5830 is it's own GPU, the first one will be device 0 and the second is device 1. You can run them in Crossfire mode while only having the monitor connected to the first one. That's the whole point of Crossfire. So you're seeing no signal on one of your monitors because you've set up the system in a single monitor setup.
Now, if you open up Catalyst, you'll see a deal called Overdrive. Click it and make sure that Windows is seeing both cards. You'll be able to adjust them separately if everything is appropriately set up. Now when you run your miner, you need to use 2 cmd prompts and make the command line the same for both of them... except which device is being used. Use device 0 for one and device 1 for the other.
In all honesty, I'd run to Radioshack and pick up some resistors to make a dummy plug. It makes life so much easier having Crossfire turned off. Just remember to extend your monitor to the second card.