How do I copy text from another application without switching to it? I know that I can highlight text from another window without changing the focus to that application by holding down Command and selecting the text. Once I have the text selected, what are my options in regard to copying the text without changing focus to that window?

It seems that when I press Command-V, it copies text from the application which is currently in the foreground.
You can right-click the selected text and choose "Copy" from the menu that pops up. If you need to copy only a single word, you can just right-click the word without selecting it first, which works in both foreground and background apps.