Learn Hardcore Russian: На вопросы в чате я буду отвечать позже, потому что сейчас я занят. - I will answer the questions in the chat later because right now I am busy.

It is just the normal government (required preposition + case) for this verb. Using вопросам (dative) would be ungrammatical here, and о вопросах would mean something else (more like about questions, not answering them).

What case is вопросы in на вопросы, and how can I tell?

Вопросы here is accusative plural because of the preposition на used with отвечать на.
A quick way to check: with inanimate nouns like вопрос, nominative plural and accusative plural look the same (вопросы), so you identify the case from the preposition + verb pattern rather than the noun ending.

Why is it в чате and not в чат?

в + prepositional is used for location: in the chat → в чате.
в + accusative is used for direction/movement: into the chat → в чат (as in зайти в чат = to enter the chat).

Why is the future written as буду отвечать instead of one word?

That is the compound future used with the imperfective verb отвечать:

я буду отвечать = I will answer / I will be answering (in a general, process-focused way)

Imperfective verbs do not have a simple future form, so Russian uses быть (in the future) + infinitive.

Could I say я отвечу позже instead of я буду отвечать позже? What changes?

Yes, and it changes the aspect/feel:

я отвечу позже (perfective ответить) sounds like a more definite, single completion: I will answer later (I will give the answer).
я буду отвечать позже (imperfective отвечать) focuses more on the process/ongoing activity or a less “bounded” idea: I will be answering later / I’ll answer later (without stressing completion).

In many everyday contexts both are acceptable; the choice depends on whether you want to emphasize completion (отвечу) or the activity/process (буду отвечать).

Why is я included? Russian often drops pronouns—can it be omitted here?

It can be omitted, especially in casual speech, because the verb form already shows the person:

На вопросы в чате буду отвечать позже… is fine.

Including я adds clarity/emphasis (for example, contrasting with someone else who might answer now). The second я in потому что сейчас я занят can also be dropped:

…потому что сейчас занят.

Is the word order fixed? Could parts of the sentence be rearranged?

Russian word order is flexible, and rearranging usually changes emphasis (what is the topic/new information), not the basic meaning. For example:

Я буду отвечать на вопросы в чате позже… (more neutral, starting with the subject)
Позже я буду отвечать на вопросы в чате… (emphasizes later)
На вопросы в чате позже буду отвечать… (fronts the object for emphasis)

The original sentence starts with На вопросы в чате to highlight what is being talked about first: the chat questions.

Why is there a comma before потому что?

Because потому что introduces a subordinate clause giving the reason. In Russian punctuation, a subordinate clause is usually separated by a comma:

main clause: …я буду отвечать позже
reason clause: потому что сейчас я занят

What part of speech is позже, and how is it used?

Позже is an adverb meaning later. It commonly modifies the verb phrase:

буду отвечать (когда?) позже = will answer later

It can also be used comparatively in other contexts (later than something else), but here it simply indicates a later time.

Why does it say я занят and not я занятый?

Занят is the short-form adjective used predicatively (as the main description/state):

я занят = I am busy (right now / as a state)

Занятый is the long-form adjective, usually used before a noun or to describe someone more like a characteristic in context:

занятый человек = a busy person
You can sometimes say я занятый in special contexts, but it usually sounds unnatural or changes the nuance (more like defining yourself as a type of person rather than stating your current state).

How does занят change for gender and number?

It agrees with the subject:

я занят (male speaker)
я занята (female speaker)
мы заняты (we are busy)
он занят / она занята / они заняты