1) Initiative is rolled when at least two, non-allied parties (such as a monster and an adventuring party) are made aware of one another (and generally recognize each other as threats). If one party is aware of the other ahead of time, then they can act until the other group becomes aware of them. Generally, if any side decides to do something that would make their opponents aware of them (such as going into the monster's layer after buffing themselves with spells), then they may take a single standard action before initiative is rolled. This is called a surprise round.
Example: A group of heroes comes across a door in a dungeon, through which a group of rowdy drunken orcs are celebrating their most recent raid. The party was moving quietly, and the orcs did not hear them. They move away from the door, and the cleric casts a number of buff spells on the fighter and rogue. They then carefully make their way back. The orcs are still oblivious, so they all draw weapons (the fighter his sword, the rogue his bow, the wizard his staff, and the cleric his mace and holy symbol.) The surprise round begins. The fighter kicks the door open as a move action taken in place of his standard. The rogue looks quickly, identifies the warchief, and fire a single shot at him, (catching him by surprise and likely dealing Sneack Attack damage, see Rogue class for details). The wizard blasts the room with a fireball against the back wall, and the cleric uses another spell to offer the players some other form of protection. Now, once this all happens, the orcs are aware of the player characters, and initiative is rolled. The rogue rolls highest, followed by the warchief, the wizard, the 4 orcs, the fighter and then the cleric. Normal combat begins (and thus the rogue makes a full-attack with his bow, even though none of the orcs have had a chance to act.)
That's the basics, though it becomes more complicated from their. The DM is ultimately responsible for arbitrating in unclear situations (like the orc chieftan heard the heroes, but didn't act: he wanted his soldiers to learn a lesson in keeping your guard up at all times. So, does he also act in the surprise round now?)
2. FORT, REF, and WILL are all references to Fortitude, Reflex, and Will saves. These allow the monsters (as well as other NPCs and your players) to make a defensive roll against certain kinds of attacks. Fortitude is rolled against poisons, reflex against area attacks like fireballs and the like, and will saves are used to shake off mental control and manipulation.
If you haven't already, you should read this: it should help with reading the monster's stats.
3. Generally, Strength (Str), Dexterity (Dex), etc. aren't usually used directly in combat. They are mostly used to calculate the other aspects of the monster's abilities.
Example: A monster's strength plays into their attacks (usually).
Here we have a monster with a claw attack: his base attack is 0, but he has a strength of 13, giving him a +1 strength modifier. This is added to his attack roll and damage roll.
Claw +1 (1d4+1)
However, let's say someone poisons the monster, and lowers the monsters strength by 5. The monster now has a strength of 8, which gives it a -1 penalty to its attack and damage rolls.
Claw -1 (1d4-1)
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1) At the beginning of combat. It's just that if one side is surprised they wait out the first round, while those that aren't can take either an action or a move but not both.
2) The listed modifiers are predetermined calculations based on other stats, but when they come into play is when you are rolling a saving throw for the monster. For example: Fort +6, Ref +5, Will +1. If the monster is hit by a spell or effect requiring a saving throw, you roll a d20 and add that modifier, and comp[are it to the difficulty class (DC).
3) Actually you are right, the attack rolls and AC may use the bonuses of the ability scores but are pre-calculated for you. What the abilities are for basically fall into two categories. First, if the creature tries to use a skill that's not listed under its "Skills" designation use the modifier of the ability score to determine it. For example just because a creature doesn't have Perception listed under skills they can still try it, just roll a d20 and add or subtract the creature's wisdom modifier. Second, whenever you or the rules call for an "Ability Check" (like if you're kicking in the door with strength or catching a tossed object with Dexterity) roll a d20 and add that ability's modifier to get your total.
I hoped that helped, any questions or clarifications needed please ask!
1) Initiative is rolled by everyone who's going to be participating in the battle at the beginning. it doesn't need to be rolled again.
Here's how surprise round works. I don't know how they symplify it in beginner box if at all, but here's the basic idea:
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Heroes A, B and C are ambushed by enemies X Y Z, who were sneaking up on the heroes.
the heroes were already in plain sight, so XYZ see all of them.
heroes A B and C roll their perception to get a chance to notice the enemies sneaking. A and B fail, so they don't know they're being attacked. C succeeds and sees Y.
The way I do it is this, and I don't remember if it's official rules or not: C, X, Y and Z all roll initiative at once, and then play out their turns. A and B are considered surprised and do not act in this round, but they can be targeted.
Say X goes before C. Because X attacks, he must roll stealth again against C's perception because he runs the risk of revealing himself by attacking. If X fails to stay hidden, then C sees him and on his turn can now target both X and Y because he is aware they are there. But he still cannot target Z until he has a chance to notice him. It's common sense, you can't attack what you are not aware of.
After this round is over, A and B roll initiative, and the battle proceeds as normal.