Learn Hardcore Japanese: tasika kondo no kadai no simekiri ha asatte no asa no hazu da. - If I’m not mistaken, the deadline for this assignment should be the morning of the day after tomorrow.

たしか means “If I remember correctly / I’m fairly sure.” It softens the statement and signals slight uncertainty.
たしかに means “indeed / certainly,” which is assertive agreement or concession, not a hedge. It wouldn’t fit this sentence’s “from memory” nuance.
You’ll often hear the stack たしか … はずだ, which doubly softens the claim: “If I recall, it should be …”

Why are there so many の? What does each one do?

の links nouns and noun-like phrases. Break-down:

今度の課題 = “this/next time’s assignment; the upcoming assignment”
課題の締め切り = “the assignment’s deadline”
明後日の朝 = “the morning of the day after tomorrow”
朝のはず = “(it) is expected to be the morning” because はず is a noun and takes の after a noun. Chaining の like this is very common in Japanese.

Why is it 朝のはずだ and not just 朝だ?

Because はず is a noun meaning “expectation/supposed-to.” To say “X should be Y,” when Y is a noun, you use the pattern N のはずだ.

締め切りは明後日の朝だ = “The deadline is the morning of the day after tomorrow.” (confident assertion)
締め切りは明後日の朝のはずだ = “The deadline should be the morning of the day after tomorrow.” (inferred/remembered expectation)

Why is は used after 締め切り? Could I use が?

は marks the topic: “As for the deadline, (it) should be …” That’s the natural choice when you’re identifying or describing the deadline’s time.
が would mark the grammatical subject and tends to sound like you’re choosing the deadline as the thing that satisfies “is expected to be tomorrow morning,” e.g., in contrastive or corrective contexts. It’s possible but less neutral here. Typical Q→A:
Q: 締め切りはいつ？ A: 締め切りは明後日の朝のはずだ。 (topic continuity)

Is this sentence polite? How do I make it more polite?

The given sentence is plain/casual (ending in だ).

Polite: たしか、今度の課題の締め切りは明後日の朝のはずです。
Softer/more deferential: たしか、今度の課題の締め切りは明後日の朝だったと思います。 (adds “I think” for extra humility)

Does 今度 mean “this” or “next”? Is it ambiguous?

It can be ambiguous. 今度 can mean:

How do you read these words, and are the kanji forms normal?

たしか (often written 確か when it’s the adjective “certain”; in this hedging use, kana is common)
今度こんど
課題かだい
締め切りしめきり (you’ll also see 締切 in signs; both are common)
明後日あさって (formal reading みょうごにち exists but is rare in conversation)
朝あさ
はず (rare kanji 筈 exists but kana はず is standard)

Should there be a に or まで after the time? How do I say “by the morning”?

With “is”: 締め切りは明後日の朝です. Don’t add に after a time with です here.
To say “by (no later than)”: use までに.
- 提出は明後日の朝までにしてください。= “Please submit by the morning the day after tomorrow.”
Signs often use まで with a specific clock time:
- 9時まで = until/by 9:00 (deadline closes at 9). Using just 朝まで can be vague; specify a time if precision matters (e.g., 明後日の朝9時まで).

What’s the difference between はずだ and でしょう / と思う / かもしれない / 予定だ?

はずだ: “should be” based on known facts/schedule; relatively confident expectation.
でしょう/だろう: “probably,” neutral inference; softer than はずだ.
と思う: “I think”; personal opinion, can be any confidence depending on tone.
かもしれない: “might/may”; low confidence.
予定だ: “is scheduled/plan is”; objective scheduling, not the speaker’s guess. In your sentence, たしか + はずだ = fairly confident but hedged by memory.

Can I move たしか to other positions?

Yes. Common placements:

たしか、今度の課題の締め切りは… (sentence-initial)
今度の課題の締め切りは、たしか、明後日の朝のはずだ。 (after the topic) Both are natural. Don’t replace it with たしかに here; that changes the meaning to “indeed.”

Does 明後日の朝 mean “by the morning” or “in the morning”?

It means “in the morning (of the day after tomorrow),” i.e., the morning time period on that day. If you want “by the morning,” say 明後日の朝までに.

Are there other words for “deadline,” and do they change the nuance?

締め切り (締切): very common, everyday “deadline/closing.”
期限: “time limit,” broader than just submissions.
提出期限: “submission deadline,” formal/clear in academic contexts.
締切日: “due date” (the date itself). All can fit similar patterns: たしか、提出期限は明後日の朝のはずだ。 (more formal-sounding)

How do I say it if I later found out I was wrong about the deadline?

Use the past expectation:

たしか、今度の課題の締め切りは明後日の朝のはずだった。= “I was pretty sure the deadline was the morning the day after tomorrow (but it wasn’t).” You can also add 思っていた for clarity: …はずだと思っていた。