In the action editor, I see there's a stash option. Is this the way to store an action for later use in animation? If I reopoen the animation action and then click on stash again, I get an error that the action has already been stored. What is the proper way to close an animation action (so I can work on another one) for later use?
By stashing the actions we are doing little more than using the NLA as a bag to hold them, and associate them with their object. If the object is linked or appended to another blend all the actions will come too. (Similarly with many export formats) Pushing them down adds a track and action strip contributing to the animation result. Stashed actions do not.
To save an action it only needs to be stashed once.
Adding a new action to the slot in action editor automatically stashes the prior.
The stashed animations do not contribute to the animation result. (They are disabled by default), whereas those that are pushed down do.
The top shown animation in the NLA editor, is the one in the slot.. (I call the active action the one in the slot, like putting a record from your stash on the turntable. ... pushing down on the other hand is like having multiple turntables, and some kind of mixing is required. Would generally say if you are going to use NLA to animate don't have one in the slot (unless editing)
Can select a stashed animation in the NLA and hit Tab to put it in the "slot" to edit. (tweak) Shown as green in my colour scheme.
Have demonstrated editing by dragging keyframes in action editor, if it's in the slot can also use timeline, dopesheet or inserting in 3d view or over property field, or in graph editor to edit.
Use Fake User.
You actually don't need to push your action down into the NLA in order to save it, you can press on the X (close) or New (duplicate) buttons to close your current action and to create a new one, the action you've closed will be saved and you'll be able to call it back later with the dropdown menu, as long as you don't close your file. But the best option is to click on the Fake User button (shield icon), it will save your action even if it's not used by any of your object/armature. An action that has no user (meaning an action that is not currently used by any object, that is not pushed down in the NLA or that has no fake user) will be automatically lost when you'll close your file:
Once you've clicked on Fake User you'll see that an F now appears in front of this action in the dropdown list, it's the garantee that your action will be available when you'll close and reopen your file. A 0, on the contrary, means that this action has no user and will be lost when you'll close the file:
The Push Down and Stash buttons push down the selected action into the NLA, the NLA is an action sequencer that allows to combine your different actions later (open a new window then select Nonlinear Animation).
You can save this way, as Batfinger says it allows to easily link the action to the object. In that case, if you make any change, you don't need to push down again, the action in the NLA is automatically synchronized.
The problem is that if you ever want to use the NLA later you'll have a stack of tracks and strips that you may not want. If you delete a strip it will also delete the action (if this action is not shared by other object(s) and if it has no Fake User). Also, if you delete an object that has this action stashed, the action will also be lost (again, if this action is not shared by other object(s) and if it has no Fake User).