1 - Yes, each Gmail label is represented by the IMAP server as a folder, and this is how they (labels) show in Aqua Mail.
2 - By default, only the default INBOX has push enabled. Long press the account, select Options and Folders / Folder / scroll to the desired label/folder, tap it, choose to "Sync as incoming".... Finally, tap the settings icon to the right of the label/folder, and enable IMAP push.
A word of caution:
Gmail's server limits the number of simultaneous network connections to 10 per account. Each folder in push mode needs one. The device losing connectivity (i.e. an elevator, an underground parking structure) might lose existing connections and Google's server will take some time to notice and cross them out.
What I'm trying to say, 1 to 3 push folders is probably the reasonable max with Gmail.