Cat defecation/urination outside of box

Question
My male cat, age 4, has been going outside of his litter box, mainly when he is mad because I don't let him outside.  I can't let him outside because he does not stay in the fenced in yard and there is a large fox population around our neighborhood who have been seen with large prey.  He is happy on a leash and even puts his head into the leash hole - but once back inside starts up again with meowing and potty behaviors.  He only gets the leash when he is not bothering us with meowing nor has had accident recently.  
We have tried:
1.  Locking him up, and rewarding him using the litterbox with more time out of "lockdown."  This tends to work until he is completely free, then we revert back about a week later.
2.  No urinary tract infection - has been checked!
3.  One litter box on each floor of house, one box per cat (3 cats - who have always been in his life)
4.  Anti-anxiety pills
5.  Feliway in problem room(s)
6.  Carpets are all professional clean - no odors left, open area and closed area litter boxes, removed air fresheners around litter boxes, use the baking soda freshener inside litter box (recently started that), clean out all 3 boxes morning and night, completely remove litter and clean boxes every 1-2 weeks.

So . . . what are we missing?

Nothing in this house has changed!!  No furniture, additional pets, location of boxes, additional people, less people, schedules, etc.

Answer
I'm not sure, Tracy!  The best I could suggest is to try different litters.  There's one on the market called Cat Attract, which is money-back guaranteed to end litter box problems.  You can find a retailer at www.preciouscat.com.

And the other thing I would suggest is trying a different anti-anxiety medication.  Not sure what he's been on.  Prozac (fluoxetine) seems to be most effective for litter box problems, followed by Clomicalm (clomipramine).  But I also have one who responds best to Elavil (amitriptyline).  So I think I'd definitely give another one a shot.