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.
Yes and yes.
As you have practically embedded every relevant rule, I will only give the reasoning without citing the sources.
Spiked Chain Training makes it double weapon, double weapon means 2 one-handed weapons, one in each hand, and you can attach those. Embedding only works with some specific weapons, and the spiked chain is not one of those. It is not hidden, as only embedded components can be hidden.
Imagine it as a scarf, extending from one hand to the other over your neck.
Houseruling it so that it is embedded and hidden would not be overpowered however in my opinion, so talk to your DM.
Do not forget to add Dual Implement Spellcaster to your Eldrich Stike, as you make an arcane attack while wielding an implement in each hand.
There is one catch however, you have to enchant the item yourself, as a Pact Blade has to be a light blade, and Spiked Chain Training does not make Spiked Chain a light blade. You can enchant it, as you treat it as a light blade.
Best Answer
This dates from 3.5e (or earlier?) - the quote below is from the d20 SRD, and is the same in the PF SRD.
This is open to interpretation, but my opinion is that, as double weapons are already a special case that differ from the general rules for two-handed weapons, that WotC intended for them to be wieldable one-handed as this implies, as well as the explicitly stated two-handed or two-weapon methods.
However...
It appears that Paizo took a different view. There is a feat to specifically allow a character to wield a quarterstaff (a double weapon) one handed.
While the value of the feat is argued (due to the "WotC" interpretation of the double weapon rules), Paizo's official position is that you can't wield one handed without it.