Potty training if you can't crate

Hi,
Looking for some help with housebreaking my sister's dog. She rescued a little chihuahua when he was about 7 months old and he is now about 1.5 years old. He is not potty trained and since I am now here where I am with him during the day, I'd like to try to work on this. He's very timid but unless you are laying down, you cannot touch him. I am only familiar with crate training dogs so have no clue how to approach this since we cannot get him into a crate. He's very playful, runs along behind you as you walk and then when you turn, he runs away and as soon as your back is turned again, he will come up to you as you are walking. If we are laying down on the couch or in bed, he will run all over the bed, and even climb all over us and let us pet him. Even comes up to you if you are not petting him and tries to put his head under your hand, or will paw at you to get your attention and he loves to sleep curled up beside one of us. We give him lots of affection but on his terms, which is when we are laying down. He also has a problem with marking his territory and if there is anything set in the floor, such as a grocery bag, or box, he will, within a day or so, usually mark it with urine. She has another male dog who is a wonderfully trained dog. The little one will potty outside at times, but he is just as apt to come in from an outside break and crap within 5 minutes. Surely there is something that will help! Any suggestions on how we could go about starting to potty train this little guy would be greatly appreciated!