This was from a test prep book:
A: この間の本貸してもらえないかな。
B: ええ,明日持ってきてます
I'm thinking that the V てもらえない form is often used for requests.
My literal parsing of 貸して ＋ もらえない is:
'Can I not receive the favor of your lending (it) for me?'
Which then translates to
'Can you lend it to me?'
Is my translation correct? I'm not sure why there is かな at the end. I thought it is used to indicate uncertainty.
You are correct that the Verb+てもらえない（か）is used to request favors. The かな softens things a bit, and is more polite, perhaps being less assuming that the favor will be granted.
Here's is a non-comprehensive list of some てもらえない request use cases roughly ordered with the top being with more politeness. The top and middle are pretty similar and may be swapped around.
貸してもらえませんか？
貸してもらえないでしょうか？
貸してもらえないかな？
貸してもらえない？
貸してもらえないか？
貸してもらえない means:
'I cannot receive the favor of your lending it to me.'
Then the か at the end changes it into a question:
'Is it the case that I cannot receive the favor of your lending it to me?'
And な adds some subjective feeling. The whole sentence (at least without な) is syntactically not a request. It is a question. It is the pragmatics (the usage) that makes it a request. The same as in English:
Can you lend me the book?
which is syntactically a question, but can be a request within the actual usage.
Your understanding is correct. To embellish on what the person above said, the more polite forms would include:
貸していただけますか。
貸していただけませんでしょうか。
お貸しいただけないでしょうか。
お貸しいただいてもよろしいでしょうか。
お借りできませんでしょうか。
頂戴できませんでしょうか。
頂戴いただけたら幸いです。
お言葉に甘えて頂戴いたします。
(the list is practically endless)
