Aw good on you for thinking about a rescue. As for the home check, they just want to ensure your home is dog friendly, your garden's secure and you have the time to devote to a dog so you don't have to be posh
As for behavioural problems, you need to be very honest with exactly what you are looking for with the rescue, they can then match you to the right dog.
Plus the new dog will need to meet your current dogs to ensure they get on.
Good luck and let us know how you get on...xx