Gaelyn

I wanted a more standalone mod, yes, but Primitive Survival apparently does not make the fish you catch into persistent entities (so they despawn after a while).

Since I initially asked for this mod, multiple mods have been made that match what I wanted:
Aquaponics and Fish Breeding: https://mods.vintagestory.at/aquaponicsfb
Ithania Expanded Fishing: https://mods.vintagestory.at/ithaniaexpandedfishing
Catch and Release: Fish: https://mods.vintagestory.at/crfish
Fishy Bucket: https://mods.vintagestory.at/fishybucket

I'm currently using all of those mods except for Fishy Bucket, and the three of them seem to work together without any issues. The first mod adds fish breeding, the second mod adds the ability to catch live fish in traps and name them (among other features), and the third mod adds craftable aquariums that you can place fish into. All of the mods allow you to catch and transport fish without killing them.

Yes.
Ever since the update above, "a player riding an elk, who uses a translocator, teleports both player and elk."

On anything before that update, you will have to push or guide the elk through the translocator; you will not be able to take it through while riding it, but you can still take it through.

And yeah, earthquakes aren't a thing, but if you are playing with sideways instability for soil/gravel/sand, there is always a chance that a landslide triggered by you or a mob buried it. If you aren't playing with that setting on, then there shouldn't be any way that the opening could have been buried.