Yes, as long as you use them on valid targets, it doesn't matter if you are on horseback. This applies to all combat skills.
No. To increase one-handed, two-handed, archery, destruction, etc, the hits must be strike a valid target, ie, an enemy.
Only for right handed people.
You might consider Bosmer or Khajiit, the former gets +10 Archery +5 Lockpicking, Sneak and Pickpocket, the latter gets +10 Sneak, +5 Archery, Lockpicking, One-Handed and Pickpocket (amongst others). Lockpicking and One-Handed rise easily (more easily than Archery) under normal use, you wouldn't want to waste a racial bonus on these, honestly. Smithing is incredibly easy to train to 100. Again, not much point wasting a racial bonus on it.
Smithing is useful for pretty much any character build in Skyrim. Coupled with enchanting, it allows you to make decent gear from the items that you acquire throughout the game. An assassin character that knows smithing and enchanting could smith some dragonscale armour (light) and enchant it with sneak and archery/one-handed bonuses.
The Dragon Scale Armour, as it is expensive, hard to find, and is the best light armour, to keep you, the Khajiit, light on your feet and almost undetectable. You could, however enchant your Dragon Scale armour parts to give a boost to your sneaking and archery. I would also advise wearing a ring, not only to increase addition to effectiveness of archery, but also one handed for those close quarters attacks, or possibly magic, depending on your skills as a mage or a warrior.
You have to buy specific perks on the One-Handed or Two-Handed skill trees, or the Sneak tree for backstab criticals.
There isnt really a "best weapon" in Skyrim, as of yet. In Oblivion the Umbra sword was possibly the best weapon in the game, but in Skyrim due to the fact that you can level one handed and 2 handed independent of each other, one persons best weapon could be another's waste of space
Each skill point earned increases damage by a tiny amount. For example, one skill point in One-Handed increases one-handed damage by 0.5% As well as that, you can select the first perk for a type of weapon, such as the first perk for one-handed weapons. It will increase damage for swords, axes, and maces by 20%. The next level of the perk will incease it by 40%, and so on
20 damage
No, it can be executed with a blocking weapon too, such as a two-handed sword.
woman were only aloud to do archery in the games but the had to cut off one of there breast. if they were right handed the had to cut off there right breast if they were left handed they would have to cut off there left breast
There is a perk in the one-handed skill tree (about leve 60 or so) that gives a chance of decapitaion when using a one-handed statding power attack. Similiarly, there is the corresponding perk in the two-handed skill tree at the same level.