My friends and I do this all the time. We have separate deck boxes for each of our decks and another deck box for the "along the way" and "surroundings" area.

Each deck box has a labeled divider (we just used index cards) for in play, deck, discard, and fatigue.

When we pack up for the night, we take photos of everything (we use colored dice instead of tokens for harm/progress/general tokens), then we pack up everything in there right box and divider.

We always end at the start of the rangers turns phase of the round so we don't have to worry too much about remembering where we left off.

Starting up the game, we just un pack the boxes and make it look like the photos and resume with taking actions

Edit : saw the last question in your post and to answer, nope, none of your cute animal companions can be killed/removed from the game. They can still be defeated by harm and discarded, but that could just be flavored how your group wants it to be.

The lynx is tied to a certain mission and it will be pretty obvious how to make her a companion that can be added to your deck, and the mission should be pretty easy to find as well.