That part goes to the smog pump, it's a TAD(Thermactor Air Divertor) of TAS Thermactor Air Sensor, or something like that. From what I have been told it will not do anything to the driveability of your rig.
Here is another pic courtesy of Steve83. We had a thread about this not too long ago and it was decided that the problem I was having was probably vacuum related.
Whoa! Well, I'd say you either have a vacuum leak, or you've confused the hell out of your computer. To find a vacuum leak, take a Butain torch (not lit of course) and run it over your vacuum lines. When the idle picks up, that's where the leak is.
After that, I'd start puttin stuff back on until I found the problem. The EGR doesn't hinder performance, so I'd definantely put that back on. If you took off the cross over that goes from the back of your heads to where the cat used to be, then make sure you plugged the holes well. If you just left that cross over on, you may want to take it off, and then plug the holes. After that, plug the vacuum lines that connected to it, or run new lines. And, if the EVR is melted, it may be the source of the leak. I'd replace it.
Man, I think you've got a bird nest to sort through. Try the vacuum thing first, and then tell us what you came up with.
By the way, after you find any leaks, if there are any, I'd disconect the battery and let it sit for about 1/2 hour to let the computer reset.