Can a restaurant disallow someone to bring a home made cake into their establishment simply because it is home made?

Yes, they can. Aside from the fact that that they're losing potential sales, they also have to follow state and local hygiene and food safety laws that might forbid outside food. Furthermore, their insurance may also forbid outside food.

Yes, they can. In many places, no outside food is allowed, by law. That is a health regulation. Restaurants are inspected and the way they store and serve food is highly regulated. Some restaurants will allow a bakery cake in. They usually will not be allowed to place such a cake in their own kitchen or cooler, or allow it to mingle with their own food. Solutions are to order a dessert off the menu, ask if a bakery cake will be allowed, invite people home for cake, or invite the guests to another nearby place afterwards for cake and coffee, or put a birthday candle into whatever else is being served.