I have a home page under a corporate name. I used to have an email link to it but this was a magnet for spam, so I insist that people contact me through LinkedIn.
** 1. What types of things should be on there, shouldn't be on there? **
The unusual solution, in particular, is C# code for converting the old Microsoft QuickBasic MBF version of floating point to IEEE double precision. This is something I did as part of a reverse-engineering project for data that had been building up in a file system since 1989. I published it because I couldn't find it anywhere else. If you solve a problem no one else is willing to touch, wrap your name around it. It's hard to argue with original material.
Working code demonstrates coding skills, original solutions are a hint you can handle unique challenges
** 2. How should it be formatted? **
It looks to me like your formatting is just right.
** 3. Do I need a picture? **
Of a portrait photo? Probably not. Such things tend to be 'set up', and don't communicate much. Other photos of you playing in the back yard with your dog or basking on the beach with a beer aren't really communicating much more. Your School section mention 'we' - could you get a group picture of the people on your project in the lab where you did your work? This would be good.
** 4.Should my resume be a [.]pdf, embedded, or both? **
At various points in the past (particularly late 1990s) one could put just about anything on the web that hinted at programming and one would get contacts. I proved this to someone who didn't believe me - he had one semester of programming completed and was in the middle of another, and I told them people with contact him with that much alone. After some prodding, he put something out, and sure enough, with in a week he was contacted. When you save your .PDF, make sure it is 'web searchable'.
An embedded resume might be more interesting in tree-view format, where someone could 'plus' on a section and expand it to read details, or collapse it to view higher level summaries.
** 5. How much "personal stuff" compared to "professional stuff" **
Your page mentions travel: some employers are interested in people that know their way around Europe, South America, or Asia. Trips to the South Pole might be interesting but won't get you any farther than NOAA. One title says 'Languages and Skills', my apologies for believing that that use of 'Languages' would be for Spanish, French, and Chinese. A better term might be 'Development Platforms'. If you do speak something other than English, this is particularly believable if you have a section on how to install SQL Server Express on a laptop written in said language.