Learn Russian: Это нижний этаж.

Here, это is used as a standalone pronoun meaning 'this is'. When you introduce or point something out with 'This is...', you always use the fixed form это, regardless of the noun's gender. If you wanted to say 'this bottom floor' (where 'this' directly modifies the noun), you would use the masculine form: этот нижний этаж.

Why does the adjective нижний end in -ий instead of the standard -ый?

Нижний is a soft-stem adjective. Because its stem ends in a soft consonant sound, it takes soft endings in all cases. In the masculine Nominative case, this means it takes -ий instead of -ый.

Since we are learning the Genitive plural, how would I change нижний этаж to mean 'of the bottom floors'?

For the adjective, because нижний has a soft stem, its Genitive plural ending is -их (not -ых), making it нижних. For the masculine noun, этаж ends in the letter ж, so it takes the Genitive plural ending -ей, making it этажей. The full phrase is нижних этажей.

Is нижний related to other words for 'down' or 'bottom'?

Yes! It comes from the root noun низ (bottom). You will also see this root in the words вниз (downwards, indicating direction) and внизу (at the bottom, indicating location).