Do you get to add your strength modifier to improvised thrown weapons? If so, how much? Can you throw an improvised weapon with two-handed in order to get x1.5 your Strength modifier in damage?
If you can, please cite a rules source. I couldn't find one.
An improvised weapon is still considered a weapon, so normal weapon rules on handedness and the amount of your strength modifier that gets applied to damage. Obviously, there cannot be a truly comprehensive list of whether a certain item can or cannot be used with two hands or if it should be a light or one-handed improvised weapon. Those bits fall under GM adjudication. The fine folks at Paizo can't be certain what random things people will want to use as weapons. Common memes suggest using gnomes or halflings as improvised weapons. I know of a group who had a player who fought with a bucket tied to ten feet of rope. I had a player who used a chair leg as a weapon for some time. He even had it plated in mithril and later given enhancements.
The only rules regarding improvised weapons that I could find (outside of the feats that deal with them) are in the Equipment section under the Weapons sub-section. As it doesn't indicate otherwise, if your GM rules that the 2x4 you pulled out of the wall can be wielded two-handed, then you get 1.5x your strength mod when you club someone with it. If he decides that the glass shard you picked up is can only be used one-handed and would be a light weapon, you would only get 0.5x your strength mod.