Nǐ kěyǐ gēn wǒ jiè shū, míngtiān zài huán gěi wǒ.

It expresses permission or a friendly offer: “you may/can (are allowed to).”

• 可以 = permission/possibility under rules. Natural for offers.
• 能 (néng) = ability or circumstantial possibility; less about permission.
• 会 (huì) = learned skill or “will” (future). Not appropriate here.

All three can work, with nuance:

• 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.

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.

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.

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”).

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。

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).

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.

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.