I'm tired of people having the same issue when the OP hasn't posted enough information to tell if it's the same or not. Try my suggestions, and let us know if it works or doesn't. Be more specific.
Your launcher log says 64-bit OS and Java 1.7.0 - are you sure you're using Java 7u5 64-bit, or that you have 32-bit windows? I'd go ahead and update your Java, and/or remove older versions.
We need a launcher log, not the name of the log, in the code box. Use pastebin if it's too long. This is usually caused by an illegal character in your techniclauncher path somewhere.
Update your Java. Also, when that happens, you should exit, and re-enter. You'll probably find yourself on top of the area you were mining in. This is a Minecraft bug, not a Technic one.
Try downloading the server software...though if you can't figure that out, I would consider allowing someone else to host for you.
It doesn't edit your minecraft.jar or anything else. It makes its own separate folder structure and everything.
That doesn't especially matter, especially if it's the 32-bit version of Windows 7. Just run with the .jar. If you don't know how to create a batch file, download the Tekkit server software, grab the launch.bat file out of there, and change the file name inside it.
Anybody who wants help with problems like this should actually submit a bug report with the requested info. It not only provides useful information for troubleshooting, but it proves you can read.
Launcher log? Launcher version (not modpack version like you show)? Memory available and launcher setting? Texture pack you may be using, and other in-game settings that may affect memory use? All these things will be helpful.
Remove your lastlogin, launcher properties, and launcher file, redownload the launcher, and run it. That's been known to fix problems like this, though it may just be the minecraft.net servers acting up.