After a brief pause while the video loads, everyone will see the video play in the center of the hangout window, so you can laugh, interject, and scratch your chins in real time. You'll also notice that Google+ automatically mutes every-one's mics while the video is playing, then unmutes it when the video ends. That makes some sense, since you're likely soliciting opinions (or just chuckles) from your audience. But if you want to interject something during the video, click and hold the "Push to Talk" button that appears under the video, speak your piece, and then let go of the button.
You already know that the Mute Mic button lets you mute your own audio feed. But what if somebody else's audio is causing problems? For example, somebody might step away from their keyboard and not know that their IM program is making a really annoying notification sound. Muting is definitely the way to go for simple annoyances and mic problems. Anyone in a hangout has the power to mute other participants -- folks can even mute the hangout's organizer, if they want.
To mute someone, point your cursor at the person's video feed in the row of attendees near the bottom of the window, and you'll see two icons: a green microphone and a red hand. Click the green mic and you'll see a little bar at the top of the hangout window that says "Mute audio for meeting participant [name]?" Click the "Mute now" button to do just that, or Cancel if you change your mind. The person's audio will go quiet so nobody can hear him, and he'll see a quick pop-up notification that he's been muted. It's up to him to unmute himself by clicking the Unmute Mic button in his hangout window. Obviously, muting is more of a temporary fix than a way to solve audio problems. If someone starts muting people just to be annoying, you might want to block them.
The red hand icon is for reporting abuse, which is for more serious problems with a hangout participant. Unlike reporting abuse for posts and photos, though, clicking this icon gives you very specific options for reporting what the person did wrong: they shared adult or violent content, or material that could harm minors; they threatened or harassed someone; they're trying to sell something or they're spamming people; or they're violating copyright laws. If someone is intentionally being a pushy, disrespectful jerk, click the red hand icon below her video feed, turn on the radio button for the type of violation she committed, and then click Report Abuse. (If you want to document the violation, you can take a screenshot of the problem before clicking the red hand icon, and then you can upload that image before clicking Report Abuse.) Google+ moderators will then investigate and might end up booting the troublemaker from the site.
When you're done hanging out, click the Exit button in the lower right corner of the hangout window (the X icon), and then close the browser window itself.
One thing worth noting about hangouts is that people you've blocked (see Blocking, Muting, and Reporting Posts) can't join your hangouts, even if they see that you're hosting one -- which they might, if a mutual contact joins your hangout. In that way, hangouts are just like most everything else on Google+: you control who can access your stuff.
Hangouts with Extras
You might have noticed that, on the "Check your hair" screen you see before you start a hangout, there's a "Hangouts with extras" link in the midst of the other buttons and options:
"Hangout with extras" is a good name for what Google+ is offering: the same kind of video chatroom you get in a regular hangout, but with a few interesting doodads to try out. As of this writing, hangouts with extras are just an alternate, still-in-testing version of hangouts, but based on how Google typically rolls out new features, the extras will probably become part of regular hangouts in the near future.
You might choose a hangout with extras if you're organizing a more work-oriented, presentation-type meeting. One of the extras is that you can give such hangouts a name, both so you can make the topic clear (as opposed to the more freewheeling nature of standard hangouts) and so that any documents you create during the hangout can be saved and indexed (more on that in a sec).
To start a hangout with extras, click the standard "Start a hangout" button on the right side of any stream's page. Next, in the window that appears, click the "Hangouts with extras" link shown above, and then click "Try Hangouts with extras." You'll then see a slightly modified "Check your hair" screen where you can name the hangout and choose who to invite. (If you don't name your hangout, Google+ names it after the date and time you started the hangout.) Once you're all set, click "Start hangout."