Yes.

It depends on where you draw the line (heh) on "straightness" and "flatness". Some planes on gems or geodes are pretty flat, but probably not perfectly flat. Another example is a spider's web between two points. That's a pretty straight line if it's taut, but again, probably not exactly perfect.

Lines and planes in the mathematical sense are 1 and 2 dimensional. They don't have any height (and lines also no width). So they can't exist as a physical object made out of atoms as they are already 3 dimensional.

They only exist as a concept.

Short answer, depends on perspective. For example surface of perfectly still lake could be considered flat, but on macro level it follows curvature of the earth. But we still use water to level our buildings, because radius of a planet is so big. On microscopic level it's anything but flat.

Someone else mentioned spider silk danging. It's also another great example, but the same perspective clause applies. But usually crystals and some geological features tend to have flat features.

maybe somebody else pointed this out:

Light ALWAYS travels in its idea of a straight-line.

Always.

It doesn't matter whether it is bent by gravity or refraction, from its perspective, it kept going straight.

Only an "outside viewer" sees any non-straight-line-ness being done, but the outside-viewer isn't seeing the curved-space or the curved-refractive-index that the photon saw.

You really have to declare to what degree you are asking. You could take a very carefully grown crystal and define a plane based on its lattice structure. But the atoms are not all perfectly placed on the lattice once you zoom in far enough. There's even gaps between the atoms! A "plane" of carbon looks more like a net to an observer on the scale of those atoms.

Is an electron a perfect sphere? Scientists probably thought so in 1900 but now ask a physicist and they will say "No, probably not".

And yes, as others have stated, our space time is not perfectly Euclidean so that's another level of uncertainty. How do you measure the small imperfections in a Euclidean model when actual space time isn't Euclidean?

As a professor used to tell my class, there are no 0s.

No, they are mathematical constructs. Everything in nature is composed of matter and the like, so there are no perfectly straight lines or flat planes.

Even a beam of light curves and refracts as it interacts with matter and space over a long enough distance.

Depending on scale. Is the surface of the lake flat?

Once you experience true level you will never go back.

Neutrinos travel in a straight line.

Cubed pyrite is one of my favorite examples of this.

According to mathematical platonism, yes.

Otherwise we have no idea. We have some models of physics, none perfectly describing our universe. We don't know the structure of space, or the structure of time.

Even if we did: what would it mean for a line or a plane to exist? There could be equivalent descriptions of our universe, some including those as objects and some only as emergent properties.

A frozen lake.

A spider's thread when it's climbing downwards.

A lot of people talk about straightness and flatness as mathematical concepts. But I think OP means it in a technical sense, as in flat like your phone screen or straight as the edges of the screen but in nature. In this sense, flatness or straightness is defined as a finite number of measured points on a surface of which the coordinates all lie between 2 mathematicaly flat/straight parallel tolerance planes/lines. By that definition, depending on what a person would consider flat, say 0.002 mm between the planes/lines, there are definetly naturally occurring crystals that would pass that test.

Is the Higgs Field a flat plane?

Simple answer: no