None of the characters make a "better" steward; they all behave exactly the same in that role. They also still work as followers: you can still ask them to follow you again, marry them, etc.

The only thing that changes when you make a follower a steward is where they hang out when they stop following you. Like the housecarl for the appropriate hold, your steward will wander around the inside and outside of your house when idle. This means they'll be hanging around when your house inevitably gets attacked by bandits, giants, or whatever else. This brings up a couple of mostly minor considerations:

• Your steward is often the first line of defense from attacks; they'll jump into combat as soon as you get within range of the house, often before you've even noticed the enemies.
• If you pick someone other than the assigned housecarl to be steward, you'll have two people hanging around guarding your house instead of just one.
• If you pick a housecarl from another hold as your steward, they leave the original house empty. This has no effect at all on gameplay but it bothered me in a slightly OCD way.

I have found a good pattern is to pick up a mercenary, get them to the level cap, then leave them as steward. I also take the housecarl out until they hit their cap and let them go again, giving you a lvl 40 and lvl 50 character hanging around your house to help fend off bandits.

Starting the Quest

After watching this video on how to start the Hearthfire house quest, you should get a clear idea on what to do.

If this video is deleted, you don't understand it or don't want to watch it, here are some excerpts from it.

• A courier should come and give you a notice to buy a house. (Citation needed?)
• If you already are the Thane of either Falkreath, Dawnstar or Morthal, the Courier will not appear, and you can simply go to the Jarl and ask for a house. (Which you will need to buy)
• After purchasing your house, some Miscellaneous quests should pop up to guide you through house production.

Also a (probably) unreliable, but possible reason why it's not starting, can be found here (with a fix), but also shown below:

Finally solved.

I've been going through my quest log completing all the quests. I was aware that I needed to complete any quests involving "radiant" quests but after finishing this already it didn't seem to do anything.

So I continued on and came across one I could not complete. Or even click on anymore. It's the one called Collect Bounty from Skald. It wouldn't complete for some reason.

So I tried several console commands:

setObjectiveCompleted BQ03 101 1

Setting the stage to different parts:

resetquest BQ03

Shortly after doing this I was greeted by a courier who dumped 3 letters on me. Orphans and visit Falkreath for purchasing were in there. yay!

By themoon.

Obtaining Wood/Lumber

Ok. There are a couple of ways to go about this.

First of all, these are all the wood/lumber mills in Skyrim, and with their respective owners:

Owner - Location - Quest

Aeri - Anga's Mill in The Pale - Chop Wood or Delivery
Bolund - The wood mill on the northeast side of Falkreath
Gilfre - Mixwater Mill in Eastmarch - Chop Wood
Grosta - Heartwood Mill in The Rift - Chop Wood or "Flight or Fight"
Hert - Half-Moon Mill in Falkreath Hold - Chop Wood
Hod - The wood mill on the southwest side of Riverwood - Chop Wood
Kharag gro-Shurkul - Solitude Sawmill in Haafingar
Thonnir - The wood mill on the north side of Morthal - Laid to Rest

Cutting your own

So, you can goto any of the above mills, and if your relationship with the mill owner is good enough, you may come and go as you please.

You can walk up to the mill, going through the really slow animation of your character moving wood/lumber onto the saw, and then obtain some free wood/lumber for your home!

To gain "reputation" with the mill owner, follow the below excerpt.

Excerpt from here:

The following table (the one above) lists lumber mill owners who will sell sawn logs via dialogue.

The 'Quest' column indicates guaranteed quests for the lumber merchant in that row which will raise their disposition enough to allow you to cut your own logs for free.

You must receive permission via dialogue before activating the mill will yield results.
If an entry in the column is blank, there are no guaranteed quests which raise that specific lumber merchant's disposition.

However, there are other ways to raise an NPC's disposition, such as certain radiant quests for which the target can be almost any NPC in the game (e.g., Rescue Mission).

While some of these quests can still raise a lumber merchant's disposition enough for them to allow you to cut your own lumber for free, they are not listed here because you are not guaranteed to receive a specific NPC as a target for that quest.

Buying it

There are two ways to buy wood/lumber.

Via. Steward

If you own a house, and have a steward who also lives there, you can ask him/her to buy wood/lumber for you indirectly, and it will appear outside immediately.

Via. Yourself

If all else fails, just walk up to the mill owner, and ask to buy some wood/lumber for 200 gold / per 20 logs.

You may need to be in a friendly relationship to do so though. So check that you are. (Clarification needed)