Resolving the trigger being tied to non-movement action is simple. Your action happens after the triggering action. Completely counter intuitive but that is it. I am not sure why you would not simply delay and then be able to select your power on the fly...
Because that would put your initiative after the target's and allow them to, say, take an attack then move away, or take multiple attacks, action point, remove themselves as a target, all kinds of things.
Aside: Have you ever seen one of the many, many movies or tv shows where someone holds someone hostage and threatens to kill someone and is shot, pushed away, whatever _before_ their ready action goes off? Cause it's pretty standard.
So when setting your movement trigger you have to define which square or squares you want to trigger your readied action.
In practice it's a lot simpler than that, but yes, you can define it as 'moves closer to the group', 'moves away', 'crosses this line of squares', but you can also do things like 'moves into a square I can charge'
Additionally, if they do not move or move into one of your cited squares you lose your action.
You say this like it's a bad thing. Is there a better option?
Another key point of confusion is if they use a movement power then you turn your readied action into an interrupt and are able to attack them before they activate their power. This is counter intuitive and seems very arbitrary in light of how all the other portions of this action work.
You aren't interrupting. Movement you're reacting to the move they just did. It doesn't matter what they do afterwards, you didn't interrupt it. For example, if someone uses Deft Strike and moves 1 square next to an enemy, and your readied 'shift back a square' triggers, then they can choose to move 1 more square to still attack. Or move somewhere else. Or use their attack on someone else other than the original target.
Additionally, this makes weird cases where the player wants to shoot the enemy before they attack but to do that they have to tie it to their movement
You don't have to - it's very effective tactic when used on melee only enemies, but a bad idea otherwise.
Conceptually what the readied action is for is to make an attack before the enemy attacks- at least that is what it is used for in many games.
So ready an action for 'If the enemy takes a hostile action'. Whether that's moving closer or readying a ranged weapon, the attack would go off.
in my opinion, it only adds to the confusion of how to adjudicate this action properly.
For what it's worth, there are a large number of people who aren't confused by the rule. You're being willfully resistant to understanding how it works because you don't like it. At least, as far as I can tell.
Can you select more than one trigger?
No. The wording is uniformly singular for picking the action or event. You can make a very broadly defined trigger, however. It's up to the DM as to how they want to limit the action, but technically you can have a trigger of 'Performs an action I dislike' or 'An enemy becomes a valid target my attack'
I would like to hear some opinions on what should the play at the table be like? How should the DM approach announcing readied actions and how should the players announce them?
Things like 'Hefts his weapon and eyes you warily' and 'Readies himself to attack' are generally clear enough I've found, but whatever is most fun for the group, as always.
What are some good alternatives to this system? Simultaneous actions, interrupt actions, simply use delay, what else?
Take powers like Disruptive Strike. Take proactive actions rather than reactive actions. Delay. Ready tactically. Use a house rule to have something like an opposed initiative roll for special cases, or just allow truly simultaneous actions where both attacks go off at once, etc.