Can you point the foot fully when there is no weight on it? If not then you have damaged the achilles tendon. Various things can pop in the ankle so the pop doesn't tell us that much. The inability to walk normally or rise onto the toes does. And this swelling without an actual ankle sprain does make one think about the achilles - perhaps a partial tear.
X-rays will not show damage to the tendon as x-rays look at bone and not connective tissue. You require a diagnostic ultrasound scan or MRI to see what's going on in the tendon. That said a good physio will be able to assess you and tell you what's going on without the expensive investigations. If they deem that the achilles is partially ruptured then they will suggest more investigations.
I suggest you get a second opinion and soon