This may be separation anxiety and you can read more about this here
http://www.wagntrain.com/SeparationAnx.htm
if you scroll down past the symptoms you can see some really good training tips to help.
For the crate training this is a good article
http://www.inch.com/~dogs/cratetraining.html
If you got your new pup at a shelter she may equate crating with the abandonment at the shelter by her first owners and also of the shelter staff leaving the dogs every night.
Sometimes in this situation you can do better with leaving the pup with the companion dog in a puppy proofed room. Or make sure the other dog is confined to the room the crate is in so the pup is not 'alone'.
If you only crate the pup when you leave then being put in the crate is a 'signal' saying 'we will be abandoning you' so you may need to use the crate for feeding meals, for short term 'rests' while you are home etc.
Do your best to tire the puppy out before you have to leave her. A tired puppy is usualy much better behaved than a pup bursting with energy.
If nothing else works you may need to talk to your vet about using a calming medication to help her be calm enough to learn you go away but you come back.
But one thing that helps instill confidence in a pup is training. Basic obedience training helps the dog learn to do what you want and it gives the pup confidence in your reliability as a leader.
You might find the free clicker training videos on the bottom of this screen interesting
http://www.clickertrainusa.com/clicker-training-videos.htm
scroll down past the ads to see the videos.
Hope this helps you!
Dogs don't really think in the manner of it being unfair that one dog is loose and another is not, though seeing the other dog doing things she'd like to join in on might trigger some effort to get out.
You may find those training tips help her even though the symptom list does not exactly match.
If she was clean in the no cage environment, then you might want to try that puppy proof room set up. Many use a baby gate to close off a kitchen for example so if an accident happens its easy to clean up. She may be far less anxious if she can just be with the other dog.
I'd suggest a 5 minute trial to see if that works for her while you walk next door and back.
Dog Expert:Rescue, Train,Breed,Care
30+ yrs dog home vet care & nursing, rescue, behavior&training, responsible show breeding, genetics