Plumbing - parts and lingo question

I had taken many of the parts into my local Ace, and they were very helpful, which is nice, but they didn't really have what I needed.

I'll be heading out to Home Depot in a few hours. I know I won't get any help, but hopefully I can muddle my way into an answer. I'll definitely be taking my parts.

I very much appreciate all the feedback here. Fingers crossed.

-Bierp