It is my understanding that objects in the Universe are not just getting farther apart but space itself is expanding and so in some real sense, higher-dimensional geometry is "real" -- if so, on a much smaller scale, can we observe for example, points whose distance is measured using all 4 (or more) dimensions? Can 4D objects like tessaracts actually be constructed? Does this expansion affect atomic-scale objects like protons and could this have an effect on their stability?
Space is expanding. But it isn't expanding into a magical higher dimensional spatial superspace. It's merely expanding into the future.
Spacetime itself is 4d and each 4d point in 4d spacetime tells you the moment and the location.
You can imagine a 4d spacetime and have the time of a 4d point be how far it is from the origin. Each time coresponds to a 3d surface in 4d, and later times corresponds to a larger 3d surface.
But there isn't a 4d space. The larger surfaces are actually later times, what you expanded into was the future (a later time) not a different space.
This is also why there isn't a "before the big bang". There isn't a point deeper than the center of the earth.
We have to have a 4d spacetime since observers disagree on how to splice the 4d spacetime in time parts and space parts. And this is because clocks and rulers give readings that depend not only on the where-when you start at and the where-when you end up at. But also on the 4d path you take, making it impossible to objectively break it into a time part and a space part.
Which means we described it as a family of surfaces based on clocks and rulers that moved along radial lines. But others will break it down differently. But we will agree on the full 4d spacetime, it's the objective thing.
Notice that all 4d points are on a radial line to the origin so no point is preferred as a center. And there is no "before the big bang" and there is no larger space, just a 4d spacetime.