For the house training I would suggest mostly what themacpack said. A strict routine is absolutely necessary. I had a male dog that liked to mark in the house. We ended up doing a lot of crate training and he is almost always under strict supervision. (Which he does not need anymore but the parents tend to freak out) When I was working with him, I plaid the two extremes. If I caught him lifting his leg he would get a smack on the hind end and a very loud growling noise and immediately taken outside. I would hang out outside until he urinated or defecated. (Sometimes this would take an hour) then I would act like he just pissed gold. Toys, treats and tons of praise. Still to this day, going potty outside gets a 'good boy'. Then he gets to come back inside for about two hours (on a leash, in sight) then back outside until he urinated then he got to come back in. And repeat all waking hours....then crated at night.
I have two cats and two dogs. Both dogs in the beginning wanted nothing to do but chase the kittens. It is a little different because they only wanted to chase but here is what I did. There must be an idea set that Cats are on the same plain as humans, they are better than the dog and should remain that way. So as you would not let your dog charge or growl at a human they should not do the same for the cat. I would invest in a squirt bottle and mix it with vinegar and water. Everytime the dog looks at the cat the dog gets squirt (avoid eyes) with the vinegar and water. Keep the dog on leash when the cats are around. With my dogs any movement towards the cats would have them flipped over on their backs faster than they could think. Any direct staring got a smack on the hind end, any growling/barking had them flipped over on their back. If they avoided eye contact or moved away from the cat it was an instant reward, I kept treats (small bits of hot dog works wonders) on me and would toss it towards the dog if they moved away. I also demand extreme respect from my dogs, if you are concerned with being bit I would not suggest trying to flip the dog over. Both of mine know the result for putting teeth at all on any human being.
Instead I would have a leash pull the dog back and 'run it over' Keep pushing it with your knees while growling "knock it off" stomp your feet. Also remember to use the squirt bottle. That has always been one of the fastest ways for sensitive dogs to not like what ever they were doing. Some will ignore it but those that hate it will stop the behavior fast. Timing (just like with horses) is very important, notice stiffness, staring, hair raised, lip twitching, lip licking....these are all signs.