To answer your first question, yes, rules for using a quarterstaff that way is already in the rules. They're under "double weapons" and "Two-handed weapons" in the PRD.

Double Weapons: Dire flails, dwarven urgroshes, gnome hooked hammers, orc double axes, quarterstaves, and two-bladed swords are double weapons. A character can fight with both ends of a double weapon as if fighting with two weapons, but he incurs all the normal attack penalties associated with two-weapon combat, just as though the character were wielding a one-handed weapon and a light weapon.

The character can also choose to use a double weapon two-handed, attacking with only one end of it. A creature wielding a double weapon in one hand can't use it as a double weapon—only one end of the weapon can be used in any given round.

and

Two-Handed: Two hands are required to use a two-handed melee weapon effectively. Apply 1-1/2 times the character's Strength bonus to damage rolls for melee attacks with such a weapon.

So yes, the core rules already provide for this.

(In fact - and I regret I don't have my rulebook here to check - I'm pretty sure that a wizard using a quartstaff is actually given as an example of this in the Player's Handbook.)

On your point 2, note that the greatsword, greatclub and other two-handed weapons you mention are weapons that require two hands to wield. They require two hands because they are larger and heavier than weapons that do less damage; Their being larger and heavier is also why they do more damage, not he fact that they require multiple hands to wield. Wielding a weapon two-handed does multiply the strength bonus you apply to damage, as mentioned above, but has no effect on a weapon's damage dice.

No

Every class proficient with martial weapons is also proficient with simple weapons, so there would be no mechanical advantage to using a trident over a spear (unless in some unforeseen circumstance you needed the extra 1lb).¹

There would be a reason to use a spear (simple weapon) and not a trident (martial weapon), but not the other way around. Bards, clerics, druids, monks, rogues, and warlocks are proficient with simple weapons but not martial weapons. For them, their proficiency bonus would be an advantage of the spear over the trident.

However, these are not the only weapons with the same mechanics (other than price and weight). The following martial weapons are mechanically identical:

• battleaxes and longswords are both versatile weapons causing 1d8 (1d10 two-handed) slashing damage
• morningstars and war picks both cause 1d8 piercing damage

1. The "Variant Entertainer: Gladiator" background (PHB, p. 131) allows a player to "replace the musical instrument in your equipment package with an inexpensive but unusual weapon, such as a trident or net." RAW this only gives the trident as equipment. However, a DM could house rule that it also grants proficiency (in place of the music instrument from the Entertainer background). If allowed and taken by a sorcerer or wizard, they would have proficiency with tridents but not simple weapons (i.e, spears).