Can anyone tell me if this metal hose piece is part of the cooling system

 

Yes, considering it goes into the water pump...

That's the upper bleed screw.

 

What year is your car? Not all the plumbing is the same for all years. Sheobox has the diagrams for them:

http://shbox.com/1/93-94_hoses.jpg

http://shbox.com/1/heater_hoses_93-94.jpg

http://shbox.com/1/95-97_hoses.jpg

http://shbox.com/1/heater_hoses.jpg

You should add a "signature" with basic info about your car - year, model, engine, trans and any major mods. That way people have all the info they need to help you.