Reddit mentions of IPEVO Ziggi-HD High-Definition USB Document Camera (Discontinued and Upgraded to IPEVO V4K)

Maybe I am not getting what you are trying to do, but if you want to capture what is going on a screen the best approach is to run screen recording software (e.g., Hangouts, GoToMeeting, WebEx, Join.me).

If for some reason that is impossible, be careful with GoPro. First, make sure you get one where you can turn off the "fisheye" effect and also one that has a monitor on the back.

I think hooking up an IPEVO camera to a laptop (as a monitor/recorder) might be a good solution.
http://www.amazon.com/Ziggi-HD-High-Definition-Document-Camera-CDVU-04IP/dp/B008DBF5Z8/ref=sr_1_2?ie=UTF8&qid=1425337432&sr=8-2&keywords=Ipevo

Also the Logitech C920 records in 1080p so you should be able to enlarge the image and not loose too much clarity.
http://www.logitech.com/en-us/product/webcam-c930e-business?crid=1252

Hope that helps.

A lot of the advice I have comes from "Best Practices" which our math team members share. That way, our kids go from 6th grade to 7th to 8th with some similar structures and procedures in place.

As for not getting into a rhythm, you could have a vaguely bullet-pointed Agenda at the front. I do this daily; it looks like this:

Agenda

• Attendance, Announcements, Questions
  
  
• Review HW (workbook)
  
  
• Distributive Property (notebook)
  
  
• Worksheet (Partners/Groups)
  
  Even if the Agenda is a little off, kids have an idea where they stand, and you have a roadmap of your class activities, even if you don't tell them how much time you expect each activity to take.
  
  If your homework review is taking too long, cut out the stuff which is painfully long or otherwise feels unproductive. I found myself going over every problem, which was downright boring to some kids. What works for me is to have students check answers with each other while I walk around with a grade book or my tablet to see who completed the assignment (for my class, I give them 5 minutes tops, and they start the moment they sit down). This is a time which must not become "off-task" time for some kids -- setting this routine early in the year really helped. They learn to ask each other questions, and compare answers, and do some basic peer-mentoring/review. Then I say something like "Okay, even if you're not finished, we're going to go over the answers, so get settled." I have a projector and an iPevo document camera for the next step: I put the teacher's edition under the camera and display the answers on the board, and read them out loud quickly, not asking if they want to go over them until the end. For each different section/concept on the page, I say "does anyone need me to any of problems nine through fourteen? No?...Okay, what about fifteen through eighteen?" Remind them they'll see this on the quiz, or on a standardized test (which makes it "them vs. the problem", or "them vs. the standardized testing board", rather than "them vs. you"). Repeatedly doing HW problems which cover the same concept is a time waster. Usually one problem per section is plenty -- any more, and they can take notes, read their book, or come to extra help.
  
  If you feel like your homework check is going too long, you can also set expectations for your students; tell them up front that you'll only be spending 10 or 12 minutes going over the homework, and stick (pretty close) to it. Cover the problems you feel most important, if you have a choice. If you don't have a document camera/projector combo, I found numbering the problems and answers on the board without doing all of them was an acceptable compromise.
  
  I never grade homework for right/wrong -- the sheer volume and time it would take (to collect, to organize, to grade, to hand back...) is too impractical. Besides, I'm not so organized as I would need to be to do it. Our kids have a workbook which really doubles as their textbook. The way I keep them honest is with a weekly quiz. I tell them which pages and which days of material it will cover, so they know what to study in their notes. Instead, I don't collect the homework; I grade their homework on how complete it is -- did they do all the problems? Did they show their work in places they would need? My department head suggests letting kids do work in their heads, but only if they can demonstrate their ability to get right answers that way. I find it takes time for me to get to know the kids before I can tell who really is able to do it in their heads, and whether any student is simply making their homework look completed for credit.
  
  I hope you find some worth in my suggestions; I realize we all teach very differently, and my methods might not fit your style at all. Best of luck with it. Let me know what works for you, if anything!
  
  -onwardknave
  
  edit: formatting bulleted list