No, for three reasons:
First, Rapier specifically states:
You can’t wield a rapier in two hands in order to apply 1½ times your Strength bonus to damage.
Second, Weapon Finesse allows you to use DEX mod instead of STR mod on attack rolls. Wielding a one handed weapon in two hands allows you to add to your damage rolls. Two completely different cases.
Third, you cannot use Weapon Finesse when wielding two handed weapons, or wielding a one handed weapon in two hands. There are a couple of specific items that change this ( iirc, Elven Courtblade from Races of the Wild and the cheesechain Spiked Chain ), but it is limited to those items.
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.
Best Answer
If your Dex modifier replaces Str for damage, it is subject to all the same modifiers
1.5x for two-handed, 0.5x for off-hand.
Anything that lets you apply your Dexterity modifier to your damage roll is deviating from the core rules, as is the prerogative of class features, feats, etc. You follow the core rules until and unless your Dex-to-damage effect tells you different.
Consider the Swordlord's Deft Strike:
Emphasis mine. The class feature has to clarify that it does not 1-1/2 times to two-handed because otherwise it would - by replacing the Str mod it is subject to all the same rules and conditions.
However, additional Dex-to-damage is not modified
Consider the 11th-level Swashbuckler deed Bleeding Wound:
This deals your Dex modifier as additional damage, not in place of the Str modifier. It is therefore not subject to the same rules, conditions, and modifiers.