It's more used more often in written English than in spoken, so it depends on how you are going to use it. I wouldn't say "I'm expanding my family" if I was talking to another person. I'd probably say "We are having a baby" or "We are adding to our family." However in a book, such as a novel, you might hear the expression "expanding the family." So, not common in spoken English, but you would be understood.

I think so. It's perfectly correct and does not sound odd. I think "grew" is more natural. Let's see what a Google search on Project Gutenberg turns up: "if the family expanded we might build two or three extra cottages." It's not common. I see several rather different usages, talking about sociology, "the family expanded into the tribe." Yes, "grew" is better: "As the family grew larger and the moans grew smaller, many a pinch came on the household..." "In the course of time when the family grew larger the task of educating them grew more arduous." "As her family grew bigger, Henrietta Hen took them on longer strolls..."