My dad surprised me today when he said he signed my up for guitar lessons. I've been playing self-taught for like 6 months. Now I don't know what to really do when I go in for the lesson. Do I start over from square one? I don't want to have a ten-minute awkward silence. Any kind of rough idea on how the lesson might go down?

Yep, he will ask you to paly something then he will get an idea of what you know and suggest some things to work on or a song that might build certain techniques you lack
Well usually in the first lesson they usually talk more then play. To see what you need to learn.
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They'll probably ask you what music you're into, what you know (or rather what you don't), and you'll go from there.
Noice!! Well, I'm only going to be in the lessons for a month, (due to vacation later :]) So should I have everything prepared, say, like:

"I've been playing for about six months. I like a wide variety of songs. I know most open chords and a little of music theory."

.. except less blunt.
being a guitar teacher myself.its so much easier if you just relax and just be cool.like if he asks you what kind of music you are into dont freak out and try to impress him(im not saying you will do this im just speaking for past expierience)and as far as the actual lessons go do everything he tells you to do no matter how stupid or pointless it is he is trying to help you.if he teached guitar right he will teach you how to read music----LET HIM----it may seem pointless but trust me if you ever want to play as a job even if its for a club band or something,more than likely you will have to read.just allow you self to be taught and you will learn
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Just out of curiosity, what if my teacher is not the kind of teacher that is suited for my style? (i.e. - a fingerpicker teaches me, when I am solely interested in just chord progressions, singing, and theory) Should I ask for another teacher? (Does that work) Or should I stay open minded and try to learn something new?

It's a hundred dollars a month :l I don't wanna really try to learn something that doesn't interest me.
Especially since your new to the guitar, a competent teacher should be able to teach you fingerpicking as well as strumming and picking.

Also, I would suggest that you try to absorb as much as you can. Don't limit yourself to just strumming. Once you gain some experience you may become interested in other styles and it will help you to have some background.