A lot of packages that Synaptic depends on are not installed on your system. You'll need to install these first. However, installing Synaptic will not help with your proxy problem. You should solve that first, because installing packages manually is a pain (that's why package managers exist).
Make sure you've read the available resources on getting APT to work with a proxy that requires authentication, starting with How to configure proxy authentication to work with Ubuntu Software Center? If you can't get your proxy to work, then:
- If possible, seek support from your system administrators.
- Check all available documentation from your organization. Often someone somewhere has documented how to use the proxy with applications other than the major web browser, but it might take a bit of hunting down.
- You can ask on this site. We'll need to have as much information you can provide about the proxy: the name of the proxy software if you have it (it's probably displayed on the page that asks for your password), the exact settings you're using, copy-paste all prompts and error messages, etc.
Furthermore you're trying to install a package for the wrong version of Ubuntu. Synaptic 0.81.2 is the version in the upcoming Ubuntu 14.10. You have
libapt-pkg4.12 0.9.7.5, which is the version in Ubuntu 12.10. You can't mix packages from different versions of Ubuntu (well, you can try, but most of the time you'll run into dependency errors like this one).
Ubuntu 12.10 is no longer supported, and most mirrors no longer carry it. You can still obtain packages from old-releases.ubuntu.com in a pinch, but you won't get any more security updates or other bug fixes. You need to upgrade as soon as possible. Direct upgrades from 12.10 to 14.04 are not supported; normally you need to upgrade version by version or from an LTS release to another LTS release, but there's an exception: you can go 12.10 → 13.10 → 14.04.