5.7L V8

I've got a case of the "I don't know what this part is called or is for." I was going to try to clean my EGR valve since I'm getting a check engine and in the process of attempting to remove it I found two hoses with "rubber" coupling that are old and oil-corroded. I'm including pictures for them but I'm having trouble finding a parts list that gives names for things so I can get new ones. And line an idiot I broke the vacuum hose between the EGR solenoid and the valve itself so that needs to get fixed too (I don't have a picture for that but can take one if needed. Basically I need some search terms better than "That hose that connects this thing to that thing."

What is it? What's it for? Where can I get a new one?

Same here