Q: Can the Flesh Flayer in a single round attack with it's chains, grapple, and then throw or pull?

No. It attacks, does damage, then entangles. After that, it's done until its next turn: the three options it gets once an enemy is entangled all require another grapple check, which is an attack. Since it's already used a full attack action to establish the grapple, it has no more attacks during this round and has to wait until the next.