Agreed with the above replies! The easiest way to an answer is to call the local zoning office and simply ask. You can also go online and find the local ordinance, and find your answer there. It usually comes down to the definition of "family" within a single-family dwelling, which is often a certain number of non-blood related persons living together. Searching the ordinance for "blood" or "related" or "unrelated" usually gives you the answer you seek. But again, the quickest way is to pick up the phone and call.
On the same call, you can ask about the specific property you have under contract and what zoning variances have been approved for it. I've seen sellers list a SFH with "6 units" only to call the township and find out there is no approval for any of it. Always verify!
As for utilities, it is indeed best to include it in the (increased) rent. I pursue a rent-by-the-room (RBR) strategy in a few properties. When showing to perspective tenants, I often hear them say that what attracted them to my listing was the "all inclusive" price. Especially for students, interns, and other young working professionals, they don't want the added hassle of hooking up utilities. They've very attracted to the "all inclusive" model, so it is best to price it that way. Another thing to keep in mind with the RBR strategy is to always reduce the potential for roommate conflict by design. If utilities are somehow being billed to tenants, it increases the likelihood of complaints such as "John takes 3 baths a day, so his water bill should be higher than mine!" or "Jenny leaves the window open in winter, so her gas/heating bill should be higher than mine!" Thus, best to just roll it into the rental price.
One option we've been considering, especially this past winter with much higher gas and heating bills, is to include utilities up to a certain point. If the tenants go over that, then they are responsible. The idea is that the utilities are still included, but it encourages tenants to be more responsible (i.e. not leave the window open in the dead of winter). That said, it could also lead to more tenant conflict. We haven't tried it yet, but we want to test it out soon. Just an idea.
Good luck @Olivia McAleese!