How to create and send a symmetric key using GPG
First of all let me answer your specific points:


No, generating a symmetric RSA key makes no sense. RSA is not symmetric. It is a public key algorithm.
There is a HUGE difference between the public and private key. The public one can only be used to encrypt and the private one is needed for decryption.
Usually in a public key setting you keep the private key private and don't send it anywhere.


I'm afraid I have to disagree with your statement that you are pretty familiar with public key cryptography. You should probably read up on the difference between symmetric crypto and public key crypto again.

Even if you use public key crypto, you need to somehow verify that the person whose public key you are using is actually the person you want to reach. Otherwise anybody could give you their public key and MitM your connection. If you know somebody only over the internet how do you know who they are anyway?