I know "runs" is the literal translation of "løber" and it's totally valid in English, but shouldn't "flows" work just as well here?
Yes, I would translate both "the river runs through ..." and "the river flows through ..." as "floden løber (i)gennem ...".
You can also say "floden strømmer (i)gennem ...", though, but it sounds a bit contrived to my ears. Here is an example:
"Skreddene er hyppigst på steder, hvor floden strømmer igennem et landskab af sedimenter som jord, sand og grus."
The "plural" form of an adjective is used with definite nouns as well. So landet isn't a plural, but a definite noun, and hele is irregular. When hele is used with a definite noun, it does not split the noun into definite article (det/den) + noun, but instead attaches itself to the definite form of the noun (hele landet, never det hele land).
Like hele dagen → the whole day, all day.