RSA encryption with private key and decryption with a public key When using the RSA cryptosystem, does it still work if you instead encrypt with the private key and decrypt with the public key? What about in the case of using RSA for sender authentication?
Mathematically it work just fine. "Encrypt" with the private key, "decrypt" with the public key. Typically, however, we say sign with the private key and verify with the public key.

As stated in the comments, it isn't just a straight forward signing of the message $m$. Typically a hash function and padding is involved. Also, often one has a separate key pair for signing.

As you can see, there are a number of caveats, but in general you are correct, it is used for sender authentication.