Smithing isn't a bad idea for bows and arrows. (Though I believe there's a merchant that sells different kinds of arrows.)
A pure archer probably wouldn't use restoration, IMO. I think that a small melee weapon like a dagger could come handy. Archers tend to have a dagger or something similar in case an enemy comes close.
I think without a RP, this right here is really all you need. Perhaps add sneak to get one shot kills with the bow?
My very first character that I made on Skyrim was a sneaky archer.
In sneak, take stealth 1, the deadly aim perk which gives you 3x more damage using a bow in sneak mode, then up the left side to silence.
Take armsman 5 and fighting stance. You need a melee weapon in case of getting over-whelmed.
Also, take shield wall 1, power bash and deadly bash for you can bash with your bow and these 3 do work with using a bow for bashing.
Take alchemy and enchanting to enchant your own gear and weapons and to make your own potions. When taking enchanting, also take the fire and frost enchanter for those dragons, and also for other enemies that need elements to defeat. This way you got a bow for regular use, a bow for fire and a bow for frost.
In light armor, take agile defender 1, custom fit, unhindered and wind walker
Hope this helps.
This.
Practice at least a little bit of one-handed, maybe invest a couple perks to up the damage but that is it. Consider it a secondary skill.
After archery, I would say sneak would suit you very well. Consider an archer to be like a sniper class in that sense. Light armor would be optional, but prefered if you're detected you can take a few hits, but no armor and you raise your sneaking ability. (though once you get the perk, wearing armor won't affect sneaking).
I would also suggest alchemy. Poisons are great for archers as well as using a few buff abilities to up your skill in archery and sneak.
In my opinion, a good build would be something like:
Primary: Archery, Sneak, Alchemy
Secondary: Light Armor (optional), One-handed.
Extras: anything that suits your fancy for non-combat...speech, lock picking, etc.
You could use smithing to help you out too, but imo it's not needed. If anything, do some basic enchanting for rings and such instead.
Another option could be to make a non-sneaking archer and wear heavy armor.
Primary: Archery, Alchemy, Heavy Armor
Secondary: One-handed, Smithing
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As for living space, that is entirely up to you. If you have Hearthfire, going to Morthal and getting the house in the swamps would be nice or Falkreath as well. If you have no desire to get any house from the Jarl's, then maybe Hunter's Rest. Easily taken by the hunters that live there. Also from an RP perspective, Crystraldrift Cave has an old follower of Kynareth in there and can be used as a camping spot.
http://uesp.net/wiki/Skyrim:Hunter%27s_Rest
http://uesp.net/wiki/Skyrim:Crystaldrift_Cave
If you plan to have her live "outdoors" so to speak, then she can use furs for armor. Also, Falkreach offers land that you can purchase, if you have the dlc Hearthfire. A nice house can be built.
You might not even need the main hall. I love using the base house as just a place to sleep and keep stuff. Great for characters where I don't like collecting things.
Nice thing about Hearthfire. One can build as much as one wants. Yet she has to have storage and a library. She can then climb up to the roof to take in her surroundings and to kill any giants that happen to roam by.
I had no idea you could do that with a bow. I've got to try that as I love bashing.
Found that out by accident once, when I went to shoot a bandit and bashed him instead. I know that the 3 perks I listed do good for me, and gives the bow a better "bashing" ability. They actually stagger back a little more than usually. I have never tried the disarming bash, so I do not know on that one.