1.7.10 is no longer supported on this forum due to its age.
I would suggest you go to the Technic forums if you want support as the pack originates from them.
Anything you need to be called on one side or the other, but not both, is what you put in the proxy.
For the "wrong" side you can just leave the method empty so it does nothing, just make sure you are also not importing any classes from the wrong side.
If you haven't done anything try deleting the .gradle folders from your workspace and C:\Users\<Your Username>\
I'll be honest I'm taking a stab in the dark here, hoping that forcing a re-download of everything might help.
The only thing I see that I think may cause issues is the Etag things at the top since everything else appears default, how many times have you tried this?
Please make your own thread and post logs as described in the EAQ and/or my signature.
Edit: Please update to a more recent version of the game, it makes it easier on everyone.
Could you please upload a screenshot showing the settings for your launcher profile and another showing your
C:/Program Files (x86)\Minecraft\runtime\jre-x64\1.8.0_25\bin\
folder in Windows Explorer?