Yes, they have a command for compile_tabs that saves all open files to the server. I'd thought it was originally tied to super-shift-d but it no longer appears to be documented in their shortcuts list although it may still be active.
Personally, I didn't like their default key binding since it overrides the default sublime text behavior of duplicating a line. You can fix that and add your own short cut key (super-shift-s felt like the most appropriate for save all), by appending the following to your user key bindings file.
I suspect that, given your issue, you're dealing with one of two situations:
You're behind a firewall. Your options are ... get out from under the oppressive firewall! Protest! yell Attica! Attica! until your IT dept has turned off the firewall. Then use the pitchforks and torches you've magically acquired to show them to the door.
You have git mis-installed or there is a ssh key issue going on. You can identify which, by setting up your own github account, and using git via ssh to push a commit to any repo. Then, retry.
Best Answer
Take a look at mavens documentation .They provide ability to get debug log
http://mavensmate.com/Plugins/Sublime_Text/Debugging
QUICK LOG SETUP
Select MavensMate > Debug > New Quick Log or control + shift + L
Use the Trace Flag UI
Create trace flag(s) based on configuration in your project's config/.debug file.