Cats can be extremely fussy, mine will not use her litter tray more than once, even if it's just a dribble, so if I haven't noticed she will pee elsewhere, mostly under my bed, grrr.
She also stopped using it after she had been scared by a loud bang outside whilst she was in mid-flow, I eventually worked out to move it into another position, but it still took her awhile to go back to it.
Spraying is different, it's used to mark territory, you can usually tell by the smell, spraying is much stronger and lingers, maybe the cat is spraying your stuff because you are the new addition and it feels threatened.
In order to stop mine peeing elsewhere after she was frightened I had to resort to shutting her in the room with the litter tray after her meals until she had used it. Everytime she did use it I praised her lots and gave her a treat, I also banned her from my bedroom until she was back to using it properly.
Mine has access to outside, but she won't go out if it's raining or cold, or if something has scared her, she's very timid despite being quite aggressive. It is hard to get cats to adapt to certain behaviour patterns as they are not as eager to please you as dogs are, but it is possible with perseverance.
If it is spraying there are products you can buy on the market to disuade them, but you will need to apply it to the things they spray on, i.e. your stuff, lol.
Anyway, good luck!
Ninja'd by MM - but still posting.
