When I insert country borders from Natural Earth, they look broken when zoomed out. When I zoom in they look correct. Is there any way to correct this, particularly when creating a map in layout?
While you country data seems to cover the whole planet (distinguishable are Cuba to the west, tilted at 90 °, Greenland overly large at the top middle, and something like Sri Lanka to the east, in the middle), your project, or more precisely, your projects coordinate reference system (CRS) covers only a small portion of the globe. While these "restricted" CRS (like the UTM-stripes) provide high accuracy of area and length for their area of use, they are not apt at displaying features outside of it.
Trying to display a global data-set using a local CRS leads to the distortions displayed above. Either create a subset of the global data to minimize the distortions, or change the projects CRS to a global CRS. Which solution is more clever depends on the scope and tasks of your project.