
Bichons can be difficult to house train, mine still wee inside if the back door isn't open. I just live with it. They do seem to get better with age though. Our bed and the spare bed is a favorite place too. I just don't let them upstairs now as I know if they get up there they will wee on the bed. I think it is scent marking.
You should maybe take her to the vets to make rule out a urine infection.
As to keeping her white, the only way would be to bath her every 2 weeks or whenever she looks dirty, but that wouldn't do her coat much good. I use Plush Puppy www.plushpuppy.co.uk/ The herbal whitening with ginseng. Its excellent and doesn't make them itch.
I keep mine clipped short as they hate grooming and it's nicer for them and I also use a rain coat for them if its wet, one of those that zip up over the spine as it's easy to put on and stop them getting dirty underneath.
Liz