Glad you're enjoying the game.
If I use a throwing weapon example, throwing hammer, throwing axe, dagger, or throw my weapon does it dissapear or get destroyed like ammunition arrows, bolts, etc.
No. Roll 1d8. Count squares around the intended target, starting at (1) which is the square between the attacker and the target and then moving clockwise from there. The distance is up to the GM but may be up to one range increment.
Example: (T) is the Target, (A) is the attacker.
456
3T7
218
A
Can I have a shield equipped and my weapon equipped and still do a power attack? It seemed like you could not do a power attack with a shield equipped.
Yes. Though using a two-handed weapon does more damage.
It shows a horse for sale in the hero's guide, sadly it only says you can ride a horse but does not explain how to have it carry stuff, have it fight or other actions.
You will quickly run into questions that the Beginner Box will not be able to answer. That's when you can start to research the full Pathfinder game at the Paizo site or the excellent d20pfsrd fan site.
Stats for a horse. These may not exactly translate to the easier Beginner Box rules, but it should give you a start.
The horse for sale in the hero's guide is likely a light riding horse and can easily carry 250 pounds, but it likely will run before fighting.
Now a heavy horse might have a +5 (1d4+5) bite attack and 2 hooves +0 (1d6+2). It could regulary carry 300-400 pounds.
Exceeding carrying capacity would slow the horse down and eventually fatigue it.
The oil that is used as lantern fuel says it can be thrown on the ground a lit on fire to cause damage to enemies. How would this work? Do I have to be close to the enemy and drop it then light it? Do I have to prep it in advance and lead an enemy on it?
A pint of lamp oil burns for 6 hours in a common lantern or lamp. You can also use a flask of lamp oil as a splash weapon. Use the rules for alchemist's fire, except that it takes a full-round action to prepare a flask with a fuse. Once it is thrown, there is a 50% chance of the flask igniting successfully.
You can pour a pint of oil on the ground to cover an area 5 feet square, provided that the surface is smooth. If lit, the oil burns for 2 rounds and deals 1d3 points of fire damage to each creature in the area.
How does the 10 foot pole work? It says you can use it to check for traps.
Many traps are location based. Using the pole allows a character to poke around and apply pressure to areas in front of her and potentially spring a trap that she may have otherwise fallen into.
10 foot poles are handy for pulling up allies who fell into a pit-trap that they not have if they had a 10 foot pole ;)
How much can one carry? It never said a weight on the items or a weight limit.
I believe the Beginner Box takes a book-keeping-light approach and doesn't bog the game down with these details until the more advanced rules. However, if you are interested, you can see carrying capacity based on Strength here. You can also search the Pathfinder PRD for the weight of equipment.