To train a dog at this age will require patience and commitment. Two walks a day will not work. The dog needs to be taken outside about every 20 minutes and given a treat for each poo and pee. Also immediately on waking in the morning, after every nap and after eating. Do you have a fenced yard? The dog needs to be taken to the same area outside each time. Feed the cat away from the dog eating the cat food is a bad habit. Of course the cat is worried, keep them apart but let them get used to each other when someone is around let them smell each other's bedding. This will also take time.
How old are you. You say "we bought" did your mother agree first.
If you cannot look after the dog find a "no kill"kennel or foster/rescue. Your entire family need to accept the responsibility of what you have done