Just sounds really odd to me. Maybe used more in US English. I'd just say 'more often'
According to Oxford, yes: http://www.oxforddictionaries.com/us/definition/american_english/often
I agree. Whether it's in the Oxford dictionary or not, I doubt any English speaker would get this sentence right because the majority would have never heard of "oftener." I can't speak for America, Canada and Australia, but as a native Brit I can honestly say that, in thirty six years, I have never heard anyone say "oftener" and I've travelled the length and breadth of the UK. "More often" is what the majority of English speakers would say.
To me these mean different things. "The neighbors are home more often" - they're at home more often than some other people are at home. "The neighbors are more often at home" - they're at home more often than they're in some other place. I wonder if it's just me who sees this difference? And also if the Norwegian sentence makes a distinction, which I couldn't tell you.
While I get your distinction, in practice both of those sentences mean the same thing, and "The neighbors are home more often" would be followed by "than others" if that's what they meant. "... more often at home" is a slightly archaic phrasing and wouldn't be heard normally.