Yes, children can live by themselves. If the Dragonborn is married, the children will always live with the spouse.
A children's bedroom is not needed in a Hearthfire manor as long as the two small beds and the dresser between them has been built upstairs (this was not the case when Hearthfire was initally released).
Any of the other houses require a children's bedroom (purchased from the Jarl's steward).
How do I buy houses in different places other than falkreath, and where can I buy homesteads other than falkreath?
Everything is on the page: Homestead.
Let it be known that if you are wealthy, you can just build all the wings you want, hire a Steward and let them furnish the entire house room by room if you don't feel like gathering the materials to manually build everything. Haven't tested it personally so I don't know if there are bugs involved with this route or not.
Well, you can only build three wings, and there are three options for each wing (what type of room it will be) Though there ARE three homsteads, so, yes, you could build each of your three homsteads with different wings. Then you could potentially have all nine wings (three of each in three homsteads)
What I don't like is that the rooms are so specific. I would want one of my homes to be "Family" meaning a kitchen, extra bedrooms, and the library. A library and a kitchen are both East wing, and so you cannot have both in the same homstead.
As far as the supplies, you can get them all relatively easily. I'd get what I could for free before buying everything from the Steward. Extra cash can be used to.. You guessed it, build the other two homsteads!
Reading the Wiki, I've noticed there is an option to rennovate your starter house into an entryway for the main hall. If you do not choose this, does the main hall get a different entry door? And, if so, does that leave the house attatched to the main hall as a guest house?
If/when you build the main hall the only option is to connect it to the small house. You can leave the small house as-is or convert it into an entry. Converting it simply removes some of the things like the bed and firepit that don't make sense after you've added the main hall (which has a better firepit and actual bedrooms) and adds cooler decorative options.
Ah, thank you.
We built all nine options on our first two play-throughs of Hearthfire (we each did a playthrough). Our consensus is that the storeroom is useless as there is so much storage in the rest of the manor, especially in the cellar.
The armory and library tie for the second lowest value. Most of their features are duplications (the smithing facilities especially, while books can be stored in a chest or barrel). Their only value is for the visuals, but the library fails there as it is in two rooms. It would have been much more impressive if it had been in one large room, like the kitchen.
The enchanter's tower is too cramped due to the mannikins and we no longer build it as the greenhouse is so valuable.
Since the children's bedroom is no longer needed to move in adopted children, we have stopped building in two of our mansions it as well, in favor of a greenhouse.
We do build the alchemy tower as it is well designed and both floors spawn hard to get ingredients (as does the kitchen which spawns salmon roe). That also gives our manors a tower.
We feel the kitchen, with its oven and the greenhouse are the two most valuable wings and always build kitchens on all our mansions. Sometimes we do build the children's room on Lakeview Manor just to increase the number of beds. Two children, a housecarl, a steward and a bard plus the Dragonborn and spouse need six beds!
Is "We" you and your spouse :o