Do I need the bedrooms to let the spouse move in with me

Looking up "Lover's" on the in game console actually shows 2 magic effects called Lover's Comfort: CDA1F and 10D96C. This implies that while they share the same name, those two effects are considered different, so yes, they should stack with each other with no problems.

Even though there are two versions of the effect that can be seen in the console, the version that is applied from the spouse and the stone are the same (could be a bug), so currently they do not stack.

According to the article on owned houses in Skyrim, the Hearthfire plots do not count towards the number of houses owned in the game statistics. This is probably because when you buy the property, it is technically not a house, just land. It is not listed as a bug, so it should be considered normal behavior. Whether this can be "fixed" or not on a gaming console depends on the software developers.

However, your basic question, "Why can't I move my children into a Hearthfire home?" is a duplicate. See this answer here. In particular, you need to build two children's beds and two children's chests, even when you only have one child, and you need extra beds for your house carl or steward. If you don't have enough beds in a particular player-built manor it won't appear as an option.