Learn Hardcore Chinese: Nǐ kěyǐ gēn wǒ jiè shū, míngtiān zài huán gěi wǒ. - You can borrow a book from me; return it to me tomorrow.

跟 (gēn): very common and colloquial for “from (someone)” in this pattern. 你可以跟我借书.
向 (xiàng): a bit more formal/literary. 你可以向我借书.
从 (cóng): emphasizes source, often with 那里/那儿: 我从他那里借了一本书. It’s less common than 跟/向 in simple offers. Avoid 和 here; 和 means “with,” not “from,” in this structure.

Does 借 (jiè) mean “borrow” or “lend”? How do I say each?

Both, depending on the pattern:

Borrow: A 向/跟 B 借 X → A borrows X from B. Example: 你可以跟我借书.
Lend: A 把 X 借给 B → A lends X to B. Example: 我可以把书借给你. So subject and the presence of 给 tell you which side is borrowing/lending.

Why is it 还给我 (huán gěi wǒ) and not 给我还?

Both orders exist, but they’re used differently:

Default after the verb: 还给我 = “return to me.” If you include the object, use 把: 明天把书还给我.
Preposed recipient: 给我还(书/钱) also occurs in speech (often more imperative: “return it to me”).
Short form 还我(书/钱) is also fine, especially in brief commands. In your sentence (no object repeated), 还给我 is the most natural.

What exactly does 再 (zài) mean here?

Here 再 means “then/later (do it afterward),” not “again.”

明天再还 = “return it tomorrow (not now; do it then).”
Contrast: 又 (yòu) is for repetition in the past or already-occurred events; it doesn’t fit a planned future action like this. You could also say: 明天再还 or 明天然后还给我 (with 然后 for “and then”).

Why is it 明天再还 and not 再明天还?

Standard order: time word first, then 再, then the verb: 明天再还.
再明天 is not used here; to say “the day after tomorrow,” use 后天.
Think of 再 as an adverb that sticks close to the verb: 明天 + 再 + 还.

How do I pronounce 还 here, and is it the same word as “still”?

Here it’s huán (second tone) meaning “to return.”
Different from hái (second tone) meaning “still/yet.”

Return: 明天把书还 (huán)给我。
Still: 我还 (hái)在图书馆。

Why no measure word before 书? Should it be 一本书?

Chinese can omit a measure word with a generic or non-specified object. 借书 can mean “borrow (a) book(s)” in general.
Use a measure word when you want to be specific:

One book: 借一本书
This book: 借这本书 Avoid 借本书 in casual speech; 本 without 这/那/一 is a formal written usage referring to “this book” (as in academic texts).

Can I omit 给我 in the second clause?

If context already makes the recipient obvious, you can say: 明天再还.
However, 还给我 is clearer, especially if more people or places are involved.

Is the comma enough to connect the two actions? Why not use a conjunction?

Yes. Chinese commonly links sequential clauses with a comma.
You may add 然后 (rán hòu) for explicit “and then”: 你可以跟我借书，然后明天再还给我.
Avoid 和 between actions; 和 typically links nouns/NPs, not verbs/clauses.

Would adding 吧 or 一下 make it more polite?

Yes:

吧 softens it into a friendly suggestion: 你可以跟我借书，明天再还给我吧。
一下 softens the act/length: 你可以跟我借一下书，明天再还给我。 Using both is fine too.

Should I use 把 in the second clause?

You can, especially if you restate the object for clarity: 明天把书还给我.
Without 把 is also fine here because the object (书) is understood from the first clause.

Are there other natural ways to phrase the whole sentence?

你可以向我借书，明天把书还给我。
我可以把书借给你，你明天再还给我。
你可以找我借书，明天再还我。

Does 书 here mean “my book,” “a book,” or “books”? How do I disambiguate?

It’s intentionally generic. To be specific:

My book(s): 跟我借我的书 / 跟我借书 (often context implies “my”).
Exactly one: 跟我借一本书
A specific one: 跟我借这本书 or 跟我借那本书.