Cryptographic notation in X3DH Key Agreement Protocol
The $DH(PK_1, PK_2)$ function is doing the usual elliptic curve Diffie-Hellman algorithm. The secret keys involved are implicit here; of course you cannot compute the shared secret key without them!  Notice that they omit the explicit use of the secret key in the $Sig(PK,M)$ function even though it too requires a secret key.
Yes, the $F$ value is used for domain separation. The input key material is the shared secret key from the ECDH function. Part of the point of an HKDF is to make the output key material appropriately pseudorandom for cryptographic use, and the ECDH shared secret needs this transformation. Finally, the HKDF info allows you to generate multiple keys with the same key material; see the comments here for more.