Learn Hardcore Japanese: watasi ha sensei no situmon ni sugu kotaeraremasu. - I can answer the teacher's question immediately.

Yes. の is linking 先生 (teacher) and 質問 (question) in a possessive/attributive way.
先生の質問 literally means “the teacher’s question(s)” (questions belonging to / asked by the teacher).

This is the very common Noun1 + の + Noun2 pattern:

田中さんの本 = Tanaka’s book
日本の会社 = Japanese company

So here 先生の質問 is simply “the teacher’s question(s).”

Why is the particle に used after 質問 instead of を?

The verb 答える (to answer) normally takes に, not を, for what you answer to.

The basic pattern is:

X に答える = to answer X / to reply to X

So:

先生の質問に答える = answer the teacher’s question(s)

Using 質問を答える is ungrammatical or at least very unnatural. Think of に here as marking the target of your answer.

What is the role of は in 私は? Could I use が instead?

は is the topic marker. 私は sets “as for me / talking about me” as the topic of the sentence:

私は先生の質問にすぐ答えられます。
= As for me, I can answer the teacher’s questions right away.

You can say 私が先生の質問にすぐ答えられます, but it changes the nuance.

が tends to mark the subject, often with a sense of focus or contrast: “I (as opposed to someone else) am the one who can answer the teacher’s questions.”

So:

私は…: neutral “about me, here’s a fact.”
私が…: answers questions like “Who can answer?” with emphasis on I.

What exactly does すぐ mean here, and what is it modifying?

すぐ is an adverb meaning “immediately / right away / without delay.”

In this sentence it modifies 答えられます, telling us how quickly you can answer:

すぐ答えられます = “can answer right away.”

So the whole sentence is:
(I) can answer the teacher’s questions immediately.

すぐ can also mean “right nearby” in other contexts (physical closeness), but here it clearly refers to time.

What grammar form is 答えられます? How is it built?

答えられます is the polite potential form of the verb 答える (to answer).

For an ichidan (る) verb like 答える:

Take the stem: 答え
Add られる to make the potential: 答えられる = can answer
Add the polite ます: 答えられます = can answer (polite)

So 答えられます means “can answer / be able to answer” in polite speech.

Why does Japanese express “can answer” with 答えられます instead of having a separate word for “can”?

Japanese usually expresses ability / possibility by using the potential form of the verb itself, not by adding a separate word like English “can.”

So:

答えます = (I) answer / (I) will answer
答えられます = (I) can answer / am able to answer

The extra られ in the verb carries the “can / be able to” meaning. You don’t need a separate “can” word.

What’s the difference between 答えられます and 答えることができます? Don’t they both mean “can answer”?

Both mean “can answer.”

答えられます is the potential form: short and very natural in normal speech.
答えることができます is the V-plain + ことができる pattern, which also means “can do X.”

Nuance:

答えられます: everyday, natural, slightly lighter.
答えることができます: a bit more formal or deliberate, often used in very formal writing, explanations, or when you want to clearly say “have the ability/possibility to do X.”

In conversation, 答えられます is usually preferred here.

What would the sentence mean if it were 答えます instead of 答えられます?

私は先生の質問にすぐ答えます。
= “I will answer the teacher’s questions right away” / “I (always) answer them right away.”

This focuses on the action (answering), not on the ability.

私は先生の質問にすぐ答えられます。
= “I can answer the teacher’s questions right away.”

This focuses on ability / possibility — you are capable of answering them immediately.

Is 答えられます potential or passive here? I learned that られる can also make the passive.

Formally, 答えられます could be either potential or passive, but in this sentence it is clearly potential.

Passive of 答える: (X が) Y に答えられる = “X is answered by Y.”
Potential of 答える: (X は) Y に答えられる = “(X) can answer Y.”

Here we have:

私は (I) as topic
先生の質問に (to the teacher’s questions)

So 私は先生の質問にすぐ答えられます naturally reads as:
“I can answer the teacher’s questions right away.”

For a passive meaning, we’d expect the question to be the subject/topic (e.g. 先生の質問はすぐ答えられます = “The teacher’s questions can be answered quickly”). The given sentence doesn’t fit that pattern, so native speakers interpret it as potential.

What’s the difference between すぐ and すぐに? Could I say すぐに答えられます?

You can say すぐに答えられます; it’s also correct.

すぐ and すぐに both mean roughly “immediately / right away.”
すぐに can sound a little more explicitly adverb-like and sometimes slightly more formal or emphatic, but the difference is small.

In everyday speech:

すぐ行きます。
すぐに行きます。

Both are fine and mean “I’ll go right away.” Same here with すぐ(に)答えられます.

Does 先生の質問 mean one question or many questions?

Japanese nouns generally don’t mark singular or plural by themselves.

So 先生の質問 could be:

“the teacher’s question” (one)
“the teacher’s questions” (more than one)

Context decides which is intended. In English translation, you choose singular or plural depending on what fits the situation best.

Can I omit 私 in this sentence? Would 先生の質問にすぐ答えられます。 still be natural?

Yes, omitting 私 is very natural.

Japanese often drops pronouns when they’re obvious from context. If it’s already clear you are talking about yourself, 先生の質問にすぐ答えられます。 by itself is perfectly natural and probably more typical.

Including 私は can be useful:

at the start of a conversation or paragraph, to introduce yourself as the topic
when you want to contrast yourself with others (私は…, “as for me…”).

Is the word order flexible? For example, can I say 私はすぐ先生の質問に答えられます instead?

Yes, Japanese word order is relatively flexible, especially for adverbs like すぐ.

All of these are grammatical and basically mean the same:

私は先生の質問にすぐ答えられます。
私はすぐ先生の質問に答えられます。
先生の質問に私はすぐ答えられます。

Small nuances:

Placing すぐ right before 答えられます slightly highlights the “immediately” aspect of the answering.
Moving 私は or 先生の質問に can add mild emphasis or contrast, but the core meaning doesn’t change.

The original order (私は先生の質問にすぐ答えられます) is very natural and neutral.

What would the casual (non-polite) version of this sentence look like?

The plain (informal) potential form is 答えられる, without ます.

So a natural casual version is:

先生の質問にすぐ答えられる。

If you keep 私, you might say:

私は先生の質問にすぐ答えられる。

This would be used when talking to friends, family, or people of equal/lower status in a casual context.