This sounds like Alex has damage to his spinal cord, which is a soft tissue structure and so this won't show up on radiographs. He may need an MRI to fully diagnose his condition and give you an idea of whether he can be expected to improve.
Since he goes outdoors the most common cause of these symptoms is a tail pull injury or tailhead trauma.
This occurs when a cat falls and lands on his tailhead/back of the pelvis or the tail gets caught (by another animal or a car) and pulled hard enough that the caudal spinal cord is stretched. Sometimes the nerves are just stretched, but in some cases they break and when that happens the damage can be permanent. Cats may not show the full extent of their injury for several days as secondary inflammation and nerve death occur. Treatment is with steroids. My concern is that he is not responding and he is having trouble controlling his eliminations, that means prognosis for improvement is very guarded.
The other possible causes of his condition are an intervertebral disc protrusion putting pressure on his spinal cord, a blood clot, granuloma (can be inflammatory or from a parasite encysting there) or mass/tumor within the spinal cord canal which places pressure on his spinal cord.
If it hasn't been done I highly recommend that he be tested for feline leukemia and feline immunodeficiency virus which predispose to cancers, and the toxoplasmosis parasite.
I think that he likely needs to either be hospitalized and have his catheter replaced as chronic urinary retention can lead to back pressure on his kidneys and kidney failure. He may need warm water enemas to remove impacted stools too if he cannot pass them on his own. An elizabethan collar (lampshade collar) needs to be placed so he doesn't continue to remove his catheter.
If this is indeed a spinal cord injury it may take weeks to months to heal, and they can get worse before they get better as inflammation after the injury sets in. If you wish to continue treatment you need to be prepared for him needing long term nursing care and that he may never get better.
Given that you may want to discuss further testing with your veterinarian as well as referral to a veterinary neurologist for an MRI.
Best of luck with Alex, please let me know if you have any further questions.