1. They need to teach the pup to go outside - otherwise they are teaching going inside is okay and it will be tougher to 'unteach' that.
2. Using a 'reusable' sponge mop shouldn't be so bad if they are rinsing it very well with clean water.
3. Have a spray bottle of Hydrogen Peroxide handy, or make a spray bottle of very cheap vodka and spray the area you are sponge cleaning with either (not both) and go over it more than once with the clean sponge mop.
4. You can use cheap vodka to clean the whole floor but it could smell like a bar until it dries. But it cleans really well and doesn't harm the dog (because you do a rinsing swipe over the tiles at the end).
5. The older dog is undoing his house-training and needs to be trained like the puppy should be getting trained. He needs to go out a lot and praised when he soils outside, when you aren't watching the dog he needs to be managed in the house with a crate, xpen or small room. Or he should be attached to you by a leash so that you are mindful of the signs and signals he is giving so you can successfully get him out in a timely manner. It is also handy to set him up to be caught so that you can claim your territory and be very clear (but not unfair) letting him know you don't like that - usually a snappy "no potty" and a clap of your hands should suffice - then take him out. This way he learns the balance of 'You do not like it when he soils inside, you love to see him soil outside'.
But this takes a lot of diligence on the people's part and everyone needs to take turns being responsible for the dog so you aren't counting on the other person doing it because too many loopholes are created for failure.
Love Them & Lead Them,
~Elizabeth & Doug
Dog Training the Way Nature Intended