My experience is that mouthing is the worst at 4 months-ish...which is how old my new baby is too. Redirect to appropriate chew toys when she mouths/nips you, hold her muzzle and say "no bite".
That's weird about the crate--I've never experienced that. Has she seen a vet about potential problems related to that?
I would say on the training issues, stay fun and stay consistent. The "NILF"--nothing in life is free for treats, going out, coming in, food etc. also helps with the stubborness in my experience. My 4 month old has even learned which of the 4 of us in the family require obedience for treats like food and fun. He listens to me FAR more than anyone else and I started with NILF right away when we got him on Saturday.
Chris, mom to:
Tank (Cardigan Welsh Corgi)
Odin (GSD 8/28/13)
Valentine-our ragdoll cat
Two great skin kids