Can I bypass the error 3194 when downgrading from iOS 6 to 5.1.1 on a locked Verizon iPhone 4S?
Unfortunately, iOS 5.1.1 is no longer being signed by Apple.
Basically, Apple provides a digital signature to iTunes when you try to restore a firmware.  If the signature isn't provided, you're unable to restore because Apple only signs the most current version of the firmware for the device you're using.

The Dev-Team Blog says this about downgrading an iPhone 4S to 5.x:


  iPad3, i4S owners will always be able to RE-restore the current 5.x OS that's already on their device. So if you're at 5.1.1 when the window closes (and you've saved your blobs), you'll always be able to RE-restore to 5.1.1 again. This makes the 5.1.1 jailbreak a lot less fragile - you don't have to worry about messing up your install with funky extensions or getting into a boot loop, because you can always RE-restore from 5.1.1 to 5.1.1 again (or from 5.0.1 to 5.0.1 again, etc). But once you fall off the 5.x train by restoring to 6.x, you'll be stuck there until the next jailbreak.


So it looks like you're stuck on iOS 6 for now.

Regarding unlocking, your best bet is to contact Verizon.  There may be a fee required before they'll unlock your phone, if they do.