[RPG] Wielding a one handed large weapon with two hands

Well, as you were asking from a RAW answer, here is from the FAQ:

Two-Handed Weapons: What kind of action is it to remove your hand from a two-handed weapon or re-grab it with both hands?

Both are free actions. For example, a wizard wielding a quarterstaff can let go of the weapon with one hand as a free action, cast a spell as a standard action, and grasp the weapon again with that hand as a free action; this means the wizard is still able to make attacks of opportunity with the weapon (which requires using two hands).

As with any free action, the GM may decide a reasonable limit to how many times per round you can release and re-grasp the weapon (one release and re-grasp per round is fair).

—Pathfinder Design Team, 03/01/13

Note the lack of penalties when having the two-handed weapon in one hand. Now to answer your questions:

Are there any RAW that specify whether or not moving is possible with a two-handed weapon held in one hand?

Nope, movement is affected mainly by armor. A wizard is able to run with his staff in one hand, and the game makes no difference in RAW between a quarterstaff and a greatsword, they are both considered two-handed weapons.

Are there any RAW that specify whether or not you are allowed to make a ranged attack with a throwing weapon with a two handed weapon held in the off hand? Is it even possible to do that?

Yep, as in above quote. Free action to switch the greatsword to one-hand, throw weapon as a standard action, free action to grab back the greatsword. Quick Draw would be helpful to draw the throwing weapon though.

Also, this question and other sources seem to indicate that Quick Draw can't be used to switch weapons as free actions.

I’m not a huge fan of using the FAQ or developer posts on the forum as a rules-source; on numerous occasions the statements made have made little sense, contradicted the rules or previous statements, and so on. And ultimately, I don’t really care what Paizo thinks the correct answer is so much as I care what’s best for my game. For the purposes of improving my game, I also care about what the rules actually say, so I know when I have to let players know that things are working differently.

So, with that in mind, I’m going to just stick to the wording of Power Attack itself:

This bonus to damage is increased by half (+50%) if you are making an attack with a two-handed weapon, a one handed weapon using two hands, or a primary natural weapon that adds 1-1/2 times your Strength modifier on damage rolls.

Emphasis mine. A two-handed weapon, even wielded in one hand, gets this benefit from Power Attack. Maybe that’s a good ruling, may it’s not, but it’s what the rule says; if I wanted to change it, I’d explicitly note it as a houserule.

Now, when you wield a two-handed weapon in one hand, there are two possibilities. Either the rules say “you may wield a two-handed weapon using one hand“ or “you may wield a weapon that’s normally two-handed as a one-handed weapon.” In the first case, it is still a two-handed weapon: it gets the 50% extra damage from Power Attack. In the second case, it is now no longer treated as a two-handed weapon, and thus does not receive that bonus.

In my own game, I’d happily houserule that anything that lets you wield a two-handed weapon “ as a one-handed weapon” still lets you apply the damage bonus, and furthermore that this situation lets you use two-weapon fighting normally even though it might still be treated as a two-handed weapon, because these sorts of abilities are very minor, and are often outrageously overpriced. The classes that receive them, or care enough about them to take them as feats, are grossly underpowered. Since I find that such power disparities make my game less enjoyable for all involved, I’d change it in a heartbeat.