You are carrying the analogy too far. In the balloon analogy the surface of the balloon is all that exists. There is no interior. Remember that it's an analogy, there is not a correspondence between the balloon and the universe.
For egg laying animals, I would guess that the most dangerous predator might be a small mammal, such as a shrew, which could sneak into dinosaur nests and kill the eggs.
since the whole solar neighborhood (including the sun) orbits the galactic center once every 250 million years. ...
Thats the Sun's orbit.
http://curious.astro.cornell.edu/about-us/159-our-solar-system/the-sun/the-solar-system/236-are-the-planes-of-solar-systems-aligned-with-the-plane-of-the-galaxy-intermediate
The oscillation is the orbit. There's nothing keeping the sun in the galactic plane. It's orbit is inclined with respect to the plane just as, for example, Venus having an orbital inclination of 3.9 degrees from the ecliptic.
Nothing is needed to hold stars in the galactic plane. If their orbit lies within the plane, that is where they stay, unless there's some force acting upon them.
I think you are confusing an ANALOGY with theory. The balloon analogy considers the surface of the balloon as a two dimensional representation of the three dimensions of the expanding universe, and is used to illustrate scalar expansion with no center. It is neither a theory or a model.
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