I'm very glad that Roxy is allowed in the laundry room... it's comforting to know that she isn't spending her nights sleeping out in the cold and she spends more time with you.

About potty training, that's very easy to solve. Take Roxy out on-leash every 30 minutes to 1 hour, that way she has no opportunity to go potty inside. Praise her when she goes outside. Leave a few Puppy training pads or newspaper in the laundry room with her, if she continues to go in there... but I doubt she will, if you walk her very very consistently. After she gets the hang of not going potty inside, increase the gap between walks a little bit (she shouldn't be without a potty break for any more than a few hours). Make sure she gets plenty of exercise and you train and visit with her a lot, and give her stimulating toys like a peanut butter-stuffed Kong and a Buster Cube-type toy to keep her occupied.

Personally... if Roxy CAN stay indoors for several days while getting her chain fixed, I do not understand why she can't stay inside at least at night all the time. You really need to call your Mom out on this, she obviously has no legit reason. Tell her you will save up for a crate to keep Roxy in your bedroom at night (absolutely NO healthy dog will go potty in a crate), and you'll give her frequent baths and clean up, potty train her, etc. It'd even be better to have her crated or in your room during the day (with plenty of exercise, of course) than to be sitting out in the snow. I'm not criticizing, just trying to give more pointers. Just KEEP TRYING! Parents do start to give in when they know it's something you really want and you're very passionate and responsible about it.