Can you cast a spell but not actually use it (e.g. I cast shocking grasp but don't discharge it)? If so, how long would it stay "stored" in my hands?
No and Yes.
Normally, a spell's effect begins right when you finish casting it. There is no general way to store a finished spell for later use without using items.
The big exception to this are spells with a range of touch, such as shocking grasp. When you finish casting such a spell, you automatically hold it in your hand, and it is only discharged when you deliver the spell. To do so, you get a free melee touch attack in the round you cast them. (I can't find a primary source, so here's a secondary source from the Magus' Spellstrike ability)
Instead of the free melee touch attack normally allowed to deliver the spell, a magus can make one free melee attack with his weapon (at his highest base attack bonus) as part of casting this spell.
If you fail to deliver the attack (i.e. miss the touch attack), the spell is not discharged, and you can try again with regular touch attacks (Holding a spell makes those armed attacks). This is described under Holding the charge.
Touch Spells and Holding the Charge
In most cases, if you don't discharge a touch spell on the round you cast it, you can hold the charge (postpone the discharge of the spell) indefinitely. You can make touch attacks round after round until the spell is discharged. If you cast another spell, the touch spell dissipates.
Some touch spells allow you to touch multiple targets as part of the spell. You can't hold the charge of such a spell; you must touch all targets of the spell in the same round that you finish casting the spell.
So the spell stays active as long as you like, though you cannot cast another spell while holding one.
Note the exception to the exception at the end.
Yes, with certain restrictions. It's called holding a charge, and you can hold a charge until you cast another spell or until the spell is discharged into a target. (Even an invalid target, although it won't do anything useful.) Only touch-range spells can be held, and spells that allow you to touch multiple targets (e.g. mass bull's strength) can't be held.
There is a way to cast a spell, and still keep the spell slot, though I'm not sure it will be allowed into your campaign. It comes from the Kingdoms of Kalamar campaign settings "Core Rulebook 4," a 2nd party, but officially liscensed contributor content. There is a feat:
Miser with Magic [Metamagic]
Your character draws upon the minimum amount of arcane energy needed to cast a spell
Prerequisite: Spellcaster level 7+
Benefit: You can make a Spellcraft roll to retain the use of a spell after you cast it. The Spellcraft DC is 10 + twice the spells level. If you succeed, you managed to conserve enough magical energy, from this and other spells you have cast, that the spell (or spell slot for sorcerers) is not considered spent and may be used again. You may only use this ability a number of spell levels equal to the ability score modifier appropriate to the type of spells you cast (Charisma for sorcerer and bard spells, for example).
Special: If you fail the roll by 5 or more, you spend insufficient energy to cast the spell at all. You lose the spell or the spell slot, and the spell has no effect.
I'm not sure if this helps with your question or not, but I hope it does.
Edit: Sorry, I just noticed that this question was specifically for Pathfinder, in which case this will most likely not be available.