[RPG] What would be an easy-to-understand class for young children

There's a variety of dimensions to whether an activity is appropriate for children of a given age or not.

Activity Appropriateness

There's no inherent age limit for the activity of roleplaying. Kids roleplay from a very young age via "cops-and-robbers" (though nowadays it's more likely ninjas vs Transformers or something). Group imaginative play as kids is RPGs without the pedantry of dice and rules. Frankly, kids roleplay more than all adults put together. My daughter and her friends basically roleplay with Littlest Pet Shops for hours on end, just without dice or formal rules. So I don't believe there's any specific age limit or recommendation for the pure act of roleplaying.

Content Appropriateness

You should check out the content of a given RPG and see what ages you think it's appropriate for. Something like D&D may not be content appropriate for certain ages in some people's minds (violence, magic). There's other games with even lighter content (Toon, the cartoon RPG, or other games aimed at children). So there it depends on the game.

The RPG industry is bizarrely conservative, however, so most RPGs don't go past what I would consider a Y7 show on TV (with some exceptions like oWoD, but even those don't meet TV-MA level). The cartoon "Adventure Time" is basically D&D, they even use D&D terms in it. So many games are pretty kid friendly, unless you're of the "Harry Potter is evil" contingent, and even then there's other games that probably omit whatever specific content you object to. So content wise most games are OK (unless you have specific requirements around that different from day time cartoons').

Ability Appropriateness

When Hasbro puts an age rating on D&D they're not likely doing it out of content concerns, as all the content in D&D is way less objectionable than an average episode of "Billy and Mandy" - they're probably doing it to indicate what age a kid likely can perform the activities required, like they have on all their board games. If you try to play Parcheesi with a 5 year old everyone will end up crying. Kids of age X probably can't do the math/read the books/understand the rules.

Of course, there are games much simpler than D&D that even younger kids could play. I made up a basic minis game I played with my daughter when she was 6, "high roll wins" on a d6 is understandable down young. If your kid is comfortable reading chapter books, they'll do OK with less complex variants of D&D and other simple games. If they're still into picture books, you'll need something lighter and maybe customized to kids. If they're reading board books, don't bother. (If you don't know what chapter, picture, and board books are you're not a parent, why are you reading this?) It's not just about the math - are they going to be disruptive; can they participate in a group doing something for a couple hours in general without flipping out? This varies by kid.

Group Appropriateness

RPGs are a group activity. What do you mean by "unsupervised?" You mean gaming with their siblings at home with you around and just not participating in the game per se, which will be fine as long as they are not so young they'll be disruptive and generate a lot of intra-sibling screaming? Or do you mean just dropping them off at the local gaming store to play "D&D or whatever" with the 15 (heck, and 25 and 35) year olds that hang out there while you go to Walmart? Or something in between? Are they going to play in a generally safe environment, and, if you're not picking them up/dropping them off, are you comfortable with their means of transportation to and from there?

RPGs are little different from any other group activity in this regard; there's a certain level of supervision and mix of kid ages and relationship distance (sibling, relative, friend, schoolmate, stranger) any parent has already decided is appropriate for their kids whether the activity at hand is playing an RPG, playing Pokemon cards, playing soccer, going to the mall, or whatever. Different people are different here and times change - when I was a kid, at seven I and whatever kids were loose on our street would go disappear into the woods nearby and pursue our own shenanigans all day; today most kids are on lockdown and aren't let out of any adult's sight for more than 5 minutes until they're 13. Whatever your deal is there, you'd apply the same standard of supervision to a group of kids playing an RPG.

Don't think an RPG group is any safer than any other group - I wouldn't send my kid to go hang out with an arbitrary group of older people without myself or an adult I trust around. Use good judgment about any group your children hang out with.

To answer your first question, yes, rules for using a quarterstaff that way is already in the rules. They're under "double weapons" and "Two-handed weapons" in the PRD.

Double Weapons: Dire flails, dwarven urgroshes, gnome hooked hammers, orc double axes, quarterstaves, and two-bladed swords are double weapons. A character can fight with both ends of a double weapon as if fighting with two weapons, but he incurs all the normal attack penalties associated with two-weapon combat, just as though the character were wielding a one-handed weapon and a light weapon.

The character can also choose to use a double weapon two-handed, attacking with only one end of it. A creature wielding a double weapon in one hand can't use it as a double weapon—only one end of the weapon can be used in any given round.

and

Two-Handed: Two hands are required to use a two-handed melee weapon effectively. Apply 1-1/2 times the character's Strength bonus to damage rolls for melee attacks with such a weapon.

So yes, the core rules already provide for this.

(In fact - and I regret I don't have my rulebook here to check - I'm pretty sure that a wizard using a quartstaff is actually given as an example of this in the Player's Handbook.)

On your point 2, note that the greatsword, greatclub and other two-handed weapons you mention are weapons that require two hands to wield. They require two hands because they are larger and heavier than weapons that do less damage; Their being larger and heavier is also why they do more damage, not he fact that they require multiple hands to wield. Wielding a weapon two-handed does multiply the strength bonus you apply to damage, as mentioned above, but has no effect on a weapon's damage dice.