They pay for everything except for pocket money.
However, as many K-pop idols have mentioned on various shows, when you are a trainee, you only get 3000 WON (roughly $4) per meal. So you would have to find food that fits within that price range.
It may seem nice that they are paying everything during your trainee years, but once you become an idol, remember the company takes 80~90% of what you make. So it's really not a good deal.
But for JYP, I remember Rain said he didn't have enough money for some meals, so he skipped a lot of meals. So maybe JYP doesn't pay for your food. I think YG treats their artists very well, but it's the hardest to get into.
You could move near the companyHQ with your family, but the company wouldn't pay for it at all, since it's your personal choice that you are moving with your family.
If you want to the comapny to pay for housing, you would be moving into their trainee dorm, with parent consent.
Remember that the company is not obliged to treat you well. Their sole responsibility is to train you, and train you hard too.