Rescued Cat - problem with litter training

I have small rental house, tenants recently moved out and Im trying take care of stray cat that was there since Spring. They been feeding him sometimes, but he never came to contact with people. I suspected he's stray, not feral so I cached him with trap and and keeping him in storage room in that house - it is small room on side of house with windows and concrete floor.
He calmed down soon and got around, he likes when I pet him and purrs, he's really nice cat, not aggressive at all. I treated him for fleas, worms, ear mites, scheduled to neuter him, and hoped I can find a good home for him. But he stayed there for 2 weeks and still not properly litter trained.
On first day I left some old pillows for him to sleep on, but next day found out he used them as litter box. He using litter box too, with regular, not clumping clay litter. But only if nothing "soft" is around. Right now I have only cardboard on floor and cardboard box for him to sleep on, nothing else. As long he stays there in this set up, he's using litter box. But when Im coming to work on repairs and letting him out, he might sneak to do it somewhere in the house.
Today he peed on mattress (it was dirty anyway and I planned to remove it) and pooped on carpet. Carpet was not hard to clean from poop, tenants left it in bad condition anyway and it will be washed later. So no significant damage so far, but as long as he does it, I dont see any future for him to ever be adopted. I really would hate to see it, he's really nice, pretty and young (about 1 y.o.) cat.

If anybody had such experience, how long it might take for cat to stop using wrong places as litter? Or if this is even possible? I dont think he ever lived in house before. Is neutering him will help?
I have three indoor only cats myself, all was rescued when they was kittens, also dog, birds, fish, and small kid, so as much as I like this cat, I cant take him in, and I would not even have place to lock him up like he is now. He also VERY scared of my dog, even seeing dog through window makes him run away and hide. But I dont have lot of time, I need finish fixing/cleaning house soon and he will not be able to stay there after that.

Hi Sansonet, welcome to Cat Site. I advise you to consult the Cat Behaviour Forum where different litter problems are discussed by several cat owners including myself.  Five years ago I adopted a 9 years old cat who had always lived in a garden. I live  in an apartment and for the first 3 days she ignored the litterbox but when we put some  pieces of her old garden poop in the litterbox and placed her on it  she immediately adopted the litterbox  and has been using it ever since without any problem. You wil find other suggestions in the Cat Behaviour Forum. Good luck. 

Thank you! Do you mean form in this site http://www.thecatsite.com/f/5/cat-behavior ?
In this case, maybe moderators can just move it?
Problem is that he knows how to use litter box, but if given option with something soft as bedding or carpet, he uses it instead.

He may be marking his territory - did the tenants have another cat?  That cat's smell may be lingering and this guy is telling everyone "this is mine now!"

Neutering will help that.

Blessings on you for stepping up to help him.  Many people wouldn't have bothered.

He's young, around one year, from my experience males starting marking close to two years when they reach full maturity. But I dont know, there lot of changes for him, he probably never was inside house in his life - first day he looked at walls and ceiling like on alien ship..
Tenants got two kittens on summer and they still was small when they moved out, those kittens often was sitting on front of house and this cat was with them - looks like they was friends. He is really gentle cat, even when I trapped him and brought him in he was scared, he never was in contact with people and always was running away, he was so terrified that he was shaking but still didn't try to bite or scratch me when I touched him
This pic was taken about 1 hour after trapping
I tried bigger litter box today (actually, storage container) and will see if it will help.