Repeating something encrypted and non-encrypted? If one wants to keep the receiver's name non encrypted, but it also appears in the encrypted message - will it leak information? (other than the receiver's name, of course.)

Let's assume a "bad" case - where there are many messages to the same person, and that the name in the encrypted message will always appear in the exact same place, and will be publicly known to be there.

Of course it doesn't matter whether it's a name, or any other information that will be present in both encrypted, and non-encrypted form.
No, it won't "leak" information, as long as you're using a modern symmetric algorithm that's resistant to known-plaintext attacks.

However, depending on the encryption mode used (and whether there's integrity checking or not), there can be other security implications, such as the data in the known spot being substituted.