Technically, you're not allowed to ready outside of combat. Although this means a lot of things that should be possible aren't really possible, so it's probably not the way a GM should run things.
Readied actions are weird.
Part of the problem is the goblin could have been walking around with his own "readied action to attack the first enemy he spots" if you could walk around with readied actions outside of combat.
From the Combat chapter of the Core Rulebook it says:
How Combat Works
Combat is cyclical; everybody acts in turn in a regular cycle of rounds. Combat follows this sequence:
1. When combat begins, all combatants roll initiative.
2. Determine which characters are aware of their opponents. These characters can act during a surprise round. If all the characters are aware of their opponents, proceed with normal rounds. See the surprise section for more information.
3. After the surprise round (if any), all combatants are ready to being the first normal round of combat.
4. Combatants act in initiative order (highest to lowest).
5. When everyone has had a turn, the next round begins with the combatant with the highest initiative, and steps 4 and 5 repeat until combat ends.
So, as soon as the goblin came close enough that the DM thought combat was ready to begin, each of you should have rolled initiative. The DM then needed to determine if the goblin was aware of you for some reason as it seems that your character was aware of the goblin. Maybe the goblin spotted the paladin when he was some way back, again, that would be up to the DM to determine.
If the goblin was unaware of the paladin, then the paladin should have had the opportunity to act during the surprise round. As you and the goblin were the only participants in the combat, you would get a single standard or move action during the surprise round.
After that, combat would proceed as normal.
To answer your questions directly:
1. Yes, if the goblin was unaware of you, then you should be able to act in a surprise round.
2. As Claxon said there really is no guidance for using a readied action out of combat, but it happens a lot. For example, we are looking at a door and one character says "if the door opens, I will attack." That could be construed as a readied action or maybe just an opportunity to act in a surprise round (each would only be a standard or move action anyway).
3. There is no half charge in the game. You could charge the goblin in a situation like this where as the rules state "If you are able to take only a standard action on your turn, you can still charge, but you are only allowed to move up to your speed (instead of up to double your speed) and you cannot draw a weapon unless you possess the Quick Draw feat. You can't use this option unless you are restricted to taking only a standard action on your turn."
4. You are both aware of each other, so if combat is imminent, the DM should have you roll initiative even if you never actually fight. Combat is more than just fighting. Combat is the entire interaction between two sides whether there is actually engaging in fighting or not. If you had a higher initiative, then you could ready and action otherwise the barbarian could declare his action(s) first.
5. As I said earlier, there is no rules for it outside of combat, so it would be up to the DM to determine outcomes like this.
6. Same as above, you asked for RAW so the answer is whoever has a higher initiative will be able to act first. Personally I would allow such a readied action in my game, but that is not the RAW you requested. I would not, however, allow you to just have a general declaration of that readied action throughout a long march with the prisoner, but if you came to a halt and were giving them water and said you were vigilant at that time, I would allow it.
I hope that long answer works for you!
Here it goes:
My paladin is behind a wall outside of a building. A goblin is coming from the other side of the building. It is not clear if he knows about me or not.
My idea was readying an attack like this: "when I ear the goblin is near, I attack him"
In my head this is like "I'm readying an action", but when the goblin is near and I was ready for the attack, my master said "roll initiative", the goblin wins, and he is acting before me.
Questions:
1. if I was concealed, wasn't I allowed to try a surprise attack? Unless the goblin though I was behind that wall. I didn't roll stealth...2. Can I ready an attack out of combat? my "readying an action" should have triggered the beginning of the combat?
3. can I prepare an "half charge"? I prepare an attack (standard action). When it is triggered, the goblin is 5 meters away. can I half charge him?
Talking about these things with my master some other doubts came out:
4. think about this situation. I see a barbarian coming form very far. I take my crossbow and point at him. I prepare the action "if he attacks me, I fire my crossbow".
he keep walking until 18 meters from me, saying he comes in peace. No combat starts until then.
Suddenly he decide to attack me.
shall we roll initiative? if he win, he can draw his sword, run 18 meters and charge while I was pointing at him with my crossbow waiting for him?5. can and, if so, in which cases, readying an action substitute a surprise round? I guess that when me and my enemy both know about each other, surprise cannot be an option. If we are one in front of the other, initiative. What about I know he must come to me passing by a door, or something like that: can I ready an action?
6. If I ready a sword to attack a prisoner if he tries to run away, if he wins the initiative, does he run away? (we are out of combat)
Readied actions are one part of the game where you have to balance usefulness with potential for abuse. In most cases, it should probably not substitute for surprise since that's pretty unfair to your target. I doubt you'd be happy if you GM always had his ambushes prepared with readied actions to shoot at the PCs as that would largely prevent you from being able to diffuse the surprise of an ambush with a good perception check. If you don't want to be subject to it, don't suggest you should be able to do it to your GM's NPCs.
That doesn't mean that I think it never makes sense to ready an action out of combat, it's just that there are times when hoping for a successful surprise is a better and more fair mechanic to use. In the case of the paladin hiding around the corner to waylay the goblin, the goblin should get his perception check and, if successful, be potentially able to foil your ambush. Maybe he spots your shadow, hears your armor scrape on the stone wall, whatever and then reacts by taking a wider route so there's significant space between him when he comes into view and you, or perhaps he just stops and waits you out. Either way, your readied action, which has to be kind of specific, gets spoiled.
As for specific questions:
1) You should have rolled your stealth check and he should have rolled perception. As I mentioned above, various things could give you away and the opposed check is supposed to reflect that. If he failed to notice you, surprise is on your side. It may still be right to roll initiative first (there are some abilities that may help him even under surprise, but chances are he doesn't have them) before any surprise actions are taken. But if surprised and lacking special abilities that protect him or let him act, he'd still not get that surprise round action if he's actually surprised.
2 & 4) As I pointed out, it's generally a bad idea. It might make sense in limited timeframes like "My buddies are searching the desk and I know there are guards about. I'm aiming my crossbow at the doorway and shooting anyone crossing the threshold." Readying actions when there's nothing special going on or no potentially imminent trigger, trying to ready an action is probably on the rules-abusive side. "I ready my crossbow to shoot anything acting hostile" while exploring a dungeon is totally abusive. "I'm pointing my crossbow at the approaching barbarian and shooting if he makes a sudden move" isn't abusive.
5) If you and an enemy both know about each other, so no surprise, readying an action shouldn't really substitute for it... unless you get initiative on him by beating his initiative roll or there's another extenuating circumstance. When do you start readying your attack to cover the doorway? When you notice his approach (and he notices your presence)? If so, that's really too late to avoid initiative because initiative will be used to determine if you manage to ready your shot before he comes in through the door. That said, if he doesn't come into your field of fire that first round, then go ahead and ready your shot. He's coming soon, the confrontation is imminent, readying should be OK. This is quite distinct from "I'm keeping watch and readying to shoot anything coming through the door in the next 8 hours or so". There's nothing really imminent there. Try keeping on a hair trigger edge for 8 hours and I'm sure you'll find it impossible.
6) This is probably better handled by other methods too. The prisoner would almost certainly suffer an AoO for trying to leave the guard's reach. The prisoner would also be best served with some kind of distraction, a bluff of some sort opposed by the guard's sense motive, to give that prisoner his opportunity to get the drop on the guard and start to flee.