My players ready actions outside of combat often, and I don't think that it ever occurred to us that it would work any other way than what I have written below.
I think this is a straightforward RAW interpretation, and it has worked out quite well for us in the past. The players only ready actions when they honestly expect enemies, and most of the time the actions never get taken, although it's pretty cool when it works out.
You have several questions there. I'll go through each of those, and one I added.
First though, let's review how readied actions work.
You take a Standard action to ready an action.
A the time you ready the action you choose the exact action and the exact trigger.
The action takes place as an immediate reaction to the trigger.
The action expires when your next turn starts.
Is it possible to be Surprised while you have a readied action?
No. At least, not normally.
The Rules Compendium says that
"If one side in a battle notices the other side without being noticed in return, it has the advantage of surprise,"
but the Dungeon Master Guide also says that
"The PCs can’t be surprised when they open a dungeon door prepared for a fight".
What I draw from this is that if the PCs are expecting a fight - they've noticed an enemy or suspect one is nearby - then they can't be surprised.
That being said, I could entertain the argument that even if they expect some enemies on the other side of the door, they could possibly be surprised by a dragon dropping from the sky. My response would be that although they are not so prepared that they have a readied action for the dragon (unless their trigger happens to include it), they are alert and prepared to defend themselves, and therefore not surprised.
When do the readied actions take place?
They take place immediately after the triggering condition, which would likely occur on an enemy's turn.
If the PC's turn comes up in the initiative order before the action's trigger occurs, then they miss the chance to perform the readied action. The Rules Compendium says
If the trigger doesn't occur or the creature chooses to ignore it, the creature can't use the readied action and instead takes its next turn as normal.
Before or after initiative is rolled and gaining any Warlord initiative bonus such as shifting?
Since this is in response to a creature's action, it is on the creature's turn, which is after initiative is rolled.
During surprise round turns? If so at the PC's initiative or at the mob's initiative?
If there is a surprise round, the PC is not surprised, and the trigger occurs during the surprise round before the PC's turn, then yes it will take place during the surprise round.
If the PC is surprised, then they cannot take any actions, including immediate reactions such as readied actions. I think this would satisfy the 'creature chooses to ignore it' clause in the above rule, and that the PC would then miss the opportunity to use their readied action.
So the PC would just take their turn on their initiative as if they had never readied an action.
Remember, readied actions don't occur on an initiative. They occur in response to a trigger.
Or do they wait to trigger until the PC is no longer Surprised?
As pointed out above, they occur once the trigger occurs, unless they expire due to the PC's turn beginning or the PC not taking the an action despite the trigger.
By the time the PC is no longer surprised an entire round has passed, and they've lost the readied action.
Also, what's the PC's initiative order after this? Can the PC wind up at a higher position than was rolled?
Immediately before the trigger of the readied action. Yes, they can.
The Rules Compendium says:
When the creature finishes the readied action, its place in the initiative order moves to directly before the creature or the event that triggered the readied action.
Usually, an action is readied in the previous turn of combat. If the action is taken, it is because the trigger occurred before their next turn, i.e. higher in the initiative order. Their new position becomes immediately before the person that just acted, meaning that they now do not get to act in this round.
So, if a character takes an action they readied before combat, they can indeed move up in the order, but they also don't get a full turn during the first round.
For how long do the actions stay readied?
Once your next turn comes up, you lose any action you readied the previous turn. We don't normally keep track of turns outside of combat, but a turn is only supposed to be a few seconds.
This means that you can't ready an action when you wake up and keep it ready all day. You ready it, and if the trigger doesn't occur within the next few seconds, then it's over.