Render a node field inside page.html.twig
After doing quite a bit more research, I was able to find enough pieces in other questions and on drupal.org to achieve what I wanted to do.

No pre-processing is needed for this answer, so no code needs to be added to your mytheme.theme file.
All of these variables are already accessible in page.html.twig by default.

To render a field's default value inside page.html.twig, you would typically use a Twig variable as in {{ node.field_some_name.value }}.

If you have an image field, and want to render the URL to the image stored in that field inside the page.html.twig template, you would use {{ file_url(node.field_some_image.entity.fileuri) }}.

Finally, if you want to render the title of a node inside the page.html.twig template, you would use {{ node.label }}.

I also learned how to use some Twig conditionals for my particular issue, but that is beyond the scope of this question/answer.