Learn Hardcore German: Die Lehrerin bittet uns, den Raum leise zu betreten. - The teacher asks us to enter the room quietly.

bitten = to request that someone do something. Pattern: jemanden bitten, etwas zu tun. Example: Die Lehrerin bittet uns, den Raum leise zu betreten.
fragen = to ask a question. Pattern: jemanden fragen + Fragewort/ob. Example: Die Lehrerin fragt uns, warum wir zu spät sind.

Here the teacher is making a request, not asking a question, so bittet is correct.

Why is it "uns" and not "wir"?

Because bitten takes the person as a direct object in the accusative case. The pronoun paradigm is:

Nominative: wir
Accusative: uns
Dative: uns

So: Die Lehrerin bittet uns, … = “The teacher asks us to …” (Accusative).

Why is it "den Raum" and not "dem Raum"?

betreten is a transitive verb that takes a direct object in the accusative. der Raum (nom.) → den Raum (acc.). So you say den Raum betreten, not dative dem Raum.
Compare: in den Raum eintreten (see below) uses a preposition and still requires accusative after in.

Why use "betreten" here? Could I say "eintreten" or "reingehen"?

betreten + Akk. = enter/step into (no preposition), a bit formal/literary: den Raum betreten. Also “step on”: den Rasen nicht betreten.
eintreten (in + Akk.) = enter (intransitive): in den Raum eintreten.
reingehen/reinkommen = colloquial “go/come in”: ins Zimmer reingehen, Komm rein!

All can fit, but your sentence is naturally formal with betreten. Everyday speech would more likely use reinkommen/eintreten.

What’s the comma doing before the zu‑infinitive? Is it required?

In modern German, the comma before an infinitive clause with zu is often optional unless introduced by um, ohne, statt, außer, als or needed to avoid ambiguity.
Both are correct here:

Die Lehrerin bittet uns, den Raum leise zu betreten.
Die Lehrerin bittet uns den Raum leise zu betreten.
The comma is common and recommended because it cleanly marks the infinitive group.

Where does "zu" go with "betreten"? How would that differ with a separable verb?

With inseparable-prefix verbs like betreten, zu goes directly in front: zu betreten.
With separable verbs, zu splits the prefix and the stem: einzutreten, aufzumachen.
Examples:

den Raum zu betreten (betreten)
ins Zimmer einzutreten (eintreten)

Is "leise" an adjective or an adverb here?

Here it’s an adverb modifying the manner of entering: leise (quietly). As an adverb, leise doesn’t change form.
As an adjective, it inflects: ein leiser Ton, leises Betreten des Raums.

Can I move "leise" around? What word order is natural?

Inside a zu‑infinitive group, complements/adjuncts usually come before the final verb. Natural:

den Raum leise zu betreten (neutral)
leise den Raum zu betreten (slight emphasis on quietness)

Avoid placing the object after the verb:

Not: … zu betreten den Raum (unnatural/wrong).

Could I use a dass‑clause instead of the zu‑infinitive?

The normal pattern is jemanden bitten, etwas zu tun. A dass‑clause is possible but usually needs the correlate darum:

Natural but heavier: Die Lehrerin bittet uns darum, dass wir den Raum leise betreten.
Without darum (…bittet uns, dass…) sounds odd to many speakers.

How does "bitten um + Akk." fit in here?

Use bitten um + Akk. when the thing requested is a noun:

Die Lehrerin bittet um Ruhe. (She asks for quiet.) For actions, prefer the zu‑infinitive:
Die Lehrerin bittet uns, den Raum leise zu betreten. A very formal nominalized version is possible but stilted: … bittet uns um leises Betreten des Raums.

Who is understood to be the subject of "betreten" in the infinitive clause?

The understood subject is the same as the object of bitten: uns. This is called “subject control”:

Die Lehrerin bittet uns, [wir] den Raum leise zu betreten. In English too: “She asks us to enter …” → “us” does the entering.

How would I negate the action (e.g., ask us not to enter)?

Place nicht before the infinitive:

Die Lehrerin bittet uns, den Raum nicht zu betreten. (not to enter the room) To negate only the manner:
… bittet uns, den Raum nicht leise zu betreten. (not quietly—i.e., loudly), though more idiomatic is to replace it: … den Raum leise zu betreten vs … den Raum laut zu betreten.

If I replace "den Raum" with a pronoun, where does it go?

Masculine accusative becomes ihn:

Die Lehrerin bittet uns, ihn leise zu betreten. Pronouns typically appear early in the clause, before adverbs like leise.

What are the past-tense forms I might hear?

Preterite of bitten: sie bat uns, den Raum leise zu betreten.
Perfekt of bitten: sie hat uns gebeten, den Raum leise zu betreten.
Preterite of betreten: wir betraten den Raum leise.
Perfekt of betreten: wir haben den Raum leise betreten.

Is there any risk of confusing "bittet" with "bietet"?

Yes—different verbs:

bitten – bittet – bat – hat gebeten = to request/ask (for an action).
bieten – bietet – bot – hat geboten = to offer.
So bittet uns = “asks us,” while bietet uns = “offers us.”

"Raum" vs. "Zimmer": which is better here?

Both can be fine, but nuance differs:

der Raum = a room/space in general; in schools you often refer to classrooms by number as Raum 203.
das Zimmer = a room in a building (bedroom, hotel room, classroom).
You could say: Die Lehrerin bittet uns, das Zimmer leise zu betreten.

Why are some words capitalized and others not?

German capitalizes all nouns: Die Lehrerin, der/die Raum, das Zimmer.
Adjectives/adverbs like leise are lower-case (unless part of certain fixed titles or at the start of a sentence).