If they kill and eat the goose, are they mysteriously hungry a few minutes later?

Number three can be called gooselighting for short.

We had a big fat goose as a quest thing in Fellowship 2e once. The city was separated into trades guilds, and in the area of those that do woodcarving, the goose snuck up on people and honked and made them cut themselves. All the fingerless woodcarvers chipped in to get the party to help.

That was a great day in that campaign. Afterwards, the overlords general and the giants attacked town and destroyed everything, and that encounter was just barely harder than the goose chase. Love that system.

That second property seems to be a great business opportunity. Unless its carcass disappears when unattended, an enterprising adventurer can just catch it, butcher it, and then expect another one to show up.

And then the butchered meat would then be cooked as desired. Peking Goose, anyone?

I also added a version of the goose into our campaign. It followed essentially the rules here and appeared in a level of the Dungeon of the Made Mage. I also gave it a "honk" action as an AoE that would require a constitution save or on fail would cause the affected creatures to stand up, drop whatever they were holding, and cover their ears, becoming restrained until the start of the gooses next turn. I think I had also given the goose a fairly weak peck attack and disengage as a bonus action.

The players would bump into the goose in heavily trapped areas, with traps that by themselves were relatively innocent. A series of chains hanging from the ceiling, but not quite touching the floor that when disturbed shot arrows at chest level towards the disturbed chain. A long hallway of difficult to spot pressure plates, that would trigger blades to swipe at anyone on the pressure plate. That kind of thing. I also created a network of goose sized twisty turvy caverns for the goose to easily navigate from trap to trap without allowing an easy line of sight for long. If the players followed through the caverns they'd be on difficult terrain leaving the goose an opportunity to break line of sight, lose them in the maze, and potentially split up the group.

You can imagine the players on their bellies sliding underneath the chains, when out of a hole a goose waddles out. The players pleading with it to be a good goose when HONK half the players standup bumping into the chain and getting shot by arrows. The goose navigating nonplused back into the small acid etched cavern with players reluctant to follow. Then there's the goose, at the end of the next hall! The players running after it as pressure plates trigger, some having the common sense to stop and take their time, others rushing with wanton abandon. The goose disappearing into another tunnel where one or two give chase losing each other and their way in the maze. The goose appears from around a cavern corner to peck a slow moving character in the face for one or two damage and disappear around the corner again.

It was one of my favorite campaign moments.

This is the perfect kind of thing to add to your game. Minimal effort on your part, players love it, and they'll come up with something awesome you can run with later.

I love it!

This is the kind of thing In love for.

The goooose!