we go in - is just as good. "Into" is seldom used. Last time I've seen "into" being used it was "break into a security system."
"We go into the large house" sounds more natural to me than "We go in the large house".
"I put the apple in the fridge", yes, but "I'm going in the house" sounds wrong to me for some reason.
Definite is only used with a direct definite object ie one with an a/az ending in a -t (or one of the direct pronouns). -ba is an indirect object.
I know this lesson is specifically about the preverbs, so obviously we want to use them in these sentences, but in everday use, as long as the actual noun that you are moving into/away from/down towards/what have you is in the sentence and also has the ending (such as in this one, were both megyu''nk and ha'z have "be/a" attached) the preverb is optional, correct? And if you aren't specifying the object, don't you also have the choice between a preverb and a separate verb (IE, I'm going in can be "en bemegyek" or "en megyek bent")? So is there ever a place where you MUST use the preverb?