Is it safe to send the keystore file unencrypted over the internet?
It is not recommended, no. Even though the important stuff in that keystore file is encrypted, if someone were to obtain it, they could bruteforce all day and night until they discovered the password that you used.
Whether or not you decide to send it over the internet is obviously up to you. I have on occasion in the past but only we small wallets used for testing that would not be missed. If this is your main wallet or contains more ETH than you are willing to lose, I would not under any circumstances send it over the internet.
If the private key is in the file, which one is it and can I simply keep a copy of the private key and restore everything from only the private key?
Geth / Mist do not give you the private key but there are ways to obtain it if you want. My cold-storage wallet is an encrypted private key on an airgapped computer, generated via MyEtherWallet.com's repo. But, I don't let that computer touch the internet, let alone the encrypted private key. I would never send that via email.
If you are using Mist/geth for your day-to-day, keeping the keystore will be the easiest and safest way to store your account access. You could hypothetically encrypt the entire keystore but again, I recommend you find another method (ie: USB drive) to move your keystore from one place to the next.
You should be also keeping your keystore files backed up in duplicate / triplicate in case anything were to happen to your computer or your house / apartment.