Learn Hardcore Japanese: ukeketotta nooto wo yomeba, hayaku siken zyunbi wo tudukerareru node ansin desu. - Reading the notes I received lets me quickly continue exam preparation, so I feel relieved.

受け取った is the past plain form of 受け取る (“to receive”).
In Japanese, verbs in plain form can directly modify a noun that follows them. Placing 受け取った before ノート creates a relative clause: “the notes that (I) received.”
Literally: 受け取ったノート = “received notes” or “the notes I received.”

How does the conditional 〜ば in 読めば work, and what nuance does it add?

読めば is the –ば conditional form of 読む. It expresses “if/when I read …”
Nuance: a direct condition leading to a result. Here it means “If I read the notes I received, then …”
Compared to other conditionals (like と or たら), ば often emphasizes a logical or expected outcome.

What does 続けられる mean here? Is it passive or potential?

続けられる comes from 続ける (“to continue”) plus the –られる ending.
In this context, it’s the potential form: “can continue.”
Passive vs. potential ambiguity: Japanese passive also uses –られる, but because it follows 試験準備を, which is the thing being continued, it must be potential: “be able to continue exam preparation.”

Why is 早く placed before 試験準備を続けられる? What does it modify?

What role does ので play in …続けられるので安心です? How is it different from から?

ので attaches to the plain form of verbs or adjectives to indicate a reason or cause.
“…できるので安心です” = “Since I can do …, I feel relieved.”
ので is softer and more objective than から, making it suitable for polite or explanatory contexts. から might feel a bit more direct or causal.

Why do we need the particle を after ノート and 試験準備? Can they be dropped?

を marks the direct object of a verb.
- ノートを読む = “to read the notes.”
- 試験準備を続ける = “to continue exam preparation.”
Dropping を would make the sentence ungrammatical or unclear, because the listener wouldn’t know what’s the object of each verb.

Is this sentence polite, casual, or somewhere in between?

Overall it’s polite due to the ending です in 安心です.
All other verbs are in their plain forms (読めば, 続けられる), which is common in written or somewhat formal but not harshly casual speech.
This mix (plain forms + です／ます) is typical for explanatory or academic contexts.

Why is the conditional clause 受け取ったノートを読めば placed at the beginning rather than at the end?

In Japanese, subordinate clauses (conditions, reasons, relative clauses) often come before the main clause.
This front-loading of background information (“if I read the notes…”) sets up the context for the main idea (“…I can continue preparing … so I’m relieved”).
It’s a natural information flow: condition → result → speaker’s feeling.