Learn Hardcore Latin: Nos in bibliotheca tacemus, ut veritatem facilius audire possimus. - We are silent in the library, so that we can hear the truth more easily.

Yes—tacemus already means we are silent / we keep quiet, so the subject we is built into the verb ending -mus.
Putting Nos first adds emphasis or contrast, like As for us, we… or We (in particular)…. It can also make the subject extra clear if the surrounding context has multiple people.

Why is it in bibliotheca and not in bibliothecam?

With in, Latin chooses case based on meaning:

in + ablative = location (in the library): in bibliothecā
in + accusative = motion toward (into the library): in bibliothecam

Here it’s about where you are being quiet, so in bibliothecā (ablative).

What case is bibliotheca, and why?

bibliothecā is ablative singular, used after in to show place where.
It’s first-declension, so:

nominative: bibliotheca
ablative: bibliothecā (often shown with a macron in learning materials)

What exactly does tacemus mean—does it mean we literally don’t speak?