Q1: This...is my boomstick feat states that a readied shotgun can be used in a melee attack as a heavy club. Would the player have untrained penalties for using it like so without the blunt weapon proficiency?
Yes, per the RAW, but as Gatac says the GC's well within his rights to negate the penalty in favor of the feat purchase.
Q2: When using AP rounds, does it only decrease the DR of an item, armor, object, etc, or does it also reduce the damage save of items, npcs, vehicles, objects, etc.
AP affects DR but not
Q3: In Chapter 5, table 5.13, size indicates for defense, attack, etc. Do players and my npcs include the bonuses for all their items, including weapons, or is that attack bonus and others used differently. An example would be lovely.
Generally, the 5.13 modifiers are applied universally, so a Large character's attacks, Defense, encumbrance, wounds, etc. would all be affected as listed. There are exceptions but they're listed with the relevant rules.
Q4: I'm having complete difficulties understanding calculating total xp reward. The whole formula of adding up the base mission xp(missions, complications, npcs), then taking that number and multiplying it by career level or TL. Could someone please make an example mission, quick and simple with different xp rewards and such so I could totally understand.
Here's a basic example of a finished mission...
Objective: Sneak into an enemy base to find the location of a captured agent (Infiltrate 5): 125 XP
Security: Motion sensor (PR 5): 40 XP
Base guards: 50 XP
Objective: Get agent back from enemy safe house (Rescue 4): 100 XP
Complication: Time Limit (get agent out before he's killed): 50 XP
Henchmen guarding agent: 110 XP and 85 XP
Objective: Get agent back to extraction point (Transport 4): 100 XP
Complication: Dramatic Scene: 0 XP
Pursuing guards (same as in Scene 1): 50 XP
* Surviving henchmen from Scene 2 may join the chase, adding their XP again here
Adding those up, we get the following base XP totals...
Scene 1: 215 XP
Scene 2: 345 XP
Scene 3: 150 XP (perhaps increased to either 235, 260, or 345 XP if either or both henchmen join in)
Total: 710 Base XP
If the mission is tackled by a group of five player characters whose Career Levels are 8, 8, 7, 7, and 2, the Threat Level is 7 ((8 + 8 + 7 + 7 + 2) divided by 5, rounded up). Multiplying base XP by 7, the characters gain the following XP totals...
Level 7 and 8 Characters: 4,970 XP
Level 2 Character: 1,420 XP
There are more variations and modifiers - action dice, TL modifiers, and such - but that's the basic structure.
Hope that helps.