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.
By default, any heavy armor is assumed to have a gauntlet attached, and nowhere is any penalty mentioned for using them with any weapon, including bows and crossbows. What people are referring to is the ability to let go of a two-handed weapon with one hand as a Free Action, so that hand can be used to make melee attacks or cast spells with somatic components. This is explicitly allowed under the rules.
Note that there is a separate item, Locked Gauntlet, that definitely does interfere with manipulating a crossbow since you cannot (easily) let go or grab an item while using it.
Bucklers, on the other hand, usuaully give a −1 penalty to attack rolls if you wear one on a hand that is being used to wield a weapon. However, bows and crossbows are exempt from this penalty. Either way, though, you also lose the AC bonus whenever you attack with that hand.
Neither of these affect reloading, however.
Best Answer
So there used to be this ambiguous rule:
As gatherer818 reports, the bolded section was removed in an errata. This is good, as it clarifies how the rule works. However, despite this statement, you could use a heavy shield two-handed even before the errata. Here’s why:
There is no such thing as an “off-hand weapon” under the general rules
The two-weapon fighting special combat option refers to “attacks made with the off hand” which is the closest the rules ever come to using the term. Even there, it is in a specific case, not a general purpose rule.
If you are not using the two-weapon fighting special combat option, you do not have an “off hand.” It is not a general term applied in the rules.
As such, the phrase “as an off-hand weapon” is descriptive
The rules text for Pathfinder is a mix of description – statements which illustrate how things are used – and proscription – rules, requirements, and limitations on actions. It can sometimes be difficult to tell the two apart; this is something that some rules systems (D&D 4E, Legend) endeavor to correct, while others (most notably rules-light systems) embrace by doing away with proscription altogether, but there are nonetheless a lot of systems, like Pathfinder, that use a mix of the two.
Since “off-hand weapon” is not a game-term, and the phrase is used in an introductory sentence for a special combat option, it is being used descriptively – how the author expects you’ll want to use it. The original author of the line, which also existed in 3.5, even explicitly stated this fact. (link pending)
Attempting to read the line proscriptively makes no sense
Both from game balance and verisimilitude perspectives, claiming that a shield can never be a primary weapon makes no sense. This has nothing to do with two-handing the shield at this point, it’s about using the shield as a one-handed weapon, rather than forcing it to be the light one (or worse, a one-handed weapon used as the off hand attack in two-weapon fighting, accruing substantial penalties)
Game Balance
Heavy shields are one-handed weapons when used to shield bash. As such, they have to be the primary weapon, or else you take large penalties in the one case where “off hand” is even mentioned in the rules – two-weapon fighting. To effectively use a heavy shield as a weapon while dual-wielding, you must be using it as your main hand, and a light weapon as your other weapon.
Verisimilitude
Which perfectly describes the combat style of the Roman legions. Seriously, the primary melee armament of Roman legionaires was a large, heavy shield, and a small, stout gladius – a short sword. The style is more effective when you have an organized formation, of course, but that doesn’t mean it suddenly become physically impossible without that. A Roman legionaire cut off from the legion and forced to fight on his own would still fight that way, because guess what – those are the weapons he’s got, and that’s the style he’s been trained in. And while it might not be ideal, it’ll still work pretty well.
Shields make fine bashing implements, particularly when they’re heavy, which they often are. Their reach leaves something to be desired, and it can make it difficult to use the shield for its primary purpose, but it still works just fine.
Two-handing the shield is therefore allowed under the rules, and also can make sense
Imagine a big, heavy shield strapped to your left arm. You can hold it up to block things, swat at guys a bit with it, but what do you do if you really want to bash someone with it, say if your primary weapon’s been knocked out of your hand?
You’d grab your left fist or wrist with your right hand, and swing with both together, putting your whole weight behind the blow. Certainly seems to justify tacking half your Strength onto the damage to me!
And I’m certain there is ample illustration of this technique from Captain America comics or movies; I’ll try to dig those up tomorrow when I get a chance. For that matter, consider Captain America: he usually fights with a shield on his arm, and no other weapon. He’ll punch with his fist, certainly, but he also fights with the shield. It’s iconic. His fighting style is often very much shield as primary weapon, fist as the off-hand, secondary weapon. Sometimes putting a little more oomph into the shield bash is worth not getting to punch.