It may be not a good idea to just through your favorite shaving brush into a travel bag. Here is a storage idea. You may want to squeeze the loft and put it into the container first to avoid any damage while pulling the brush out of the container.
Good one. I use an empty pill bottle will holes drilled in both ends to allow some air flow should the brush be still slightly damp.
Good idea. Some sort of cover similar to those toothbrush covers they make would be a great idea for brushes.
And if you forget to clean the container good, it isn't going to be like rubbing pepper spray onto your face.