Provably Secure Password Authenticated Key Exchange Based on RLWE for the Post-QuantumWorld
A password authenticated key exchange (PAKE) scheme assumes that the two parties have some shared secret (the password) with too little entropy. They now want to agree on a key and authenticate each other using this password.

The trick is to do this in such a way that an attacker cannot brute force the password with recorded data. This ensures the password remaining secure in spite of the low entropy because you always have to talk to one of the parties to try a guess.

So 1. and 2. are out of scope for PAKE schemes. The password is just assumed to be pre-shared. 3. and 4.: Yes, $\mathcal{C}, \mathcal{S}$ identify the client and the server. These must usually be included  in authentication protocols to avoid tricky relay attacks. I recommend Boyd's book for details.