Can I stealth towards a target that fails a passive (Or Active) perception check and If I hit, I crit AND THEN ROLL FOR INITIATIVE?
Reason for question is thinking about going assassin.
Short answer is Not quite.
Running general D&D rules is up to the DM when they are open to interpretation. AL doesn't step in and specify how to interpret the rules, it just prohibits home brew. Surprise in AL is the same as surprise in any D&D game.
In your stealth example, if a character can be seen or heard then they are not hidden. If your character is not behind cover, could be seen clearly by the creature, then they aren't hidden. This means that for some DMs, no, you could not sneak up on another creature UNLESS there was something that was causing a distraction that might prevent or discourage them looking in your direction. In that case, a DM might allow a perception check to see if they notice you coming but for many DMs there would have to be a reason why they would not notice you walking in the open first.
What would happen next would also be up to DM interpretation of the rules but they generally go as follows:
First, attacks should not be made outside of the intiative order. As soon as attack rolls are being made, the characters and creatures ARE in combat and are in initiative.
"During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative."
The first thing the DM decides is who is surprised. If they decide that the creature you are attacking is surprised then they do not get to take a turn on their first initiative. You will then be able to attack them before they get to attack whether they are higher in the initiative order than you or not. When surprised, a creature can not take reactions and can not take any actions or move on their first turn. After their first turn, the creature is no longer surprised.
For most characters IF your attack hits, it will only be a critical hit if you roll a 20, if you don't roll a critical hit then the attack results in a normal hit. For an assassin rogue, if the rogue's initiative is higher than a target that has not yet taken a turn in the combat then they have advantage on the attack roll. If that target is also surprised then any hit obtained by the assassin rogue will also be a critical hit as long as the target has not had a turn in the combat yet and is thus still surprised.
Attacks, in general, do not occur outside of initiative since attacks=combat encounter=initiative
To be slightly pedantic, "If they decide that the creature you are attacking is surprised then they do not get to take a turn on their first initiative" is incorrect.
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. The distinction is that after your turn ends, you are no longer surprised and can take reactions (like casting shield spell, Battlemaster Parry ability, etc).
I have had situations where a wizard was surprised, but rolled exceptionally well for AC and was before an enemy swashbuckling rogue. The rogue moved and attacked, hoping for sneak attack since they were one-on-one, but the wizard was able to react and cast Shield. This de-railed the combat for 2 minutes while I explained this ruling.
I thought I said that? Though, I could have made it more explicit.
"When surprised, a creature can not take reactions and can not take any actions or move on their first turn. After their first turn, the creature is no longer surprised."
A creature who is no longer surprised can use reactions - which happens after their first turn in the combat has passed - though I didn't state it explicitly so thanks for the clarification. I did say that surprise ends after their first turn.
Can I stealth towards a target that fails a passive (Or Active) perception check and If I hit, I crit AND THEN ROLL FOR INITIATIVE?
Reason for question is thinking about going assassin.
The Dm decides on Surprise. But I would occasionally allow what you stated.
No Gaming is Better than Bad Gaming.
Short answer is Not quite.
Running general D&D rules is up to the DM when they are open to interpretation. AL doesn't step in and specify how to interpret the rules, it just prohibits home brew. Surprise in AL is the same as surprise in any D&D game.
In your stealth example, if a character can be seen or heard then they are not hidden. If your character is not behind cover, could be seen clearly by the creature, then they aren't hidden. This means that for some DMs, no, you could not sneak up on another creature UNLESS there was something that was causing a distraction that might prevent or discourage them looking in your direction. In that case, a DM might allow a perception check to see if they notice you coming but for many DMs there would have to be a reason why they would not notice you walking in the open first.
What would happen next would also be up to DM interpretation of the rules but they generally go as follows:
First, attacks should not be made outside of the intiative order. As soon as attack rolls are being made, the characters and creatures ARE in combat and are in initiative.
"During a round, each participant in a battle takes a turn. The order of turns is determined at the beginning of a combat encounter, when everyone rolls initiative."
The first thing the DM decides is who is surprised. If they decide that the creature you are attacking is surprised then they do not get to take a turn on their first initiative. You will then be able to attack them before they get to attack whether they are higher in the initiative order than you or not. When surprised, a creature can not take reactions and can not take any actions or move on their first turn. After their first turn, the creature is no longer surprised.
For most characters IF your attack hits, it will only be a critical hit if you roll a 20, if you don't roll a critical hit then the attack results in a normal hit. For an assassin rogue, if the rogue's initiative is higher than a target that has not yet taken a turn in the combat then they have advantage on the attack roll. If that target is also surprised then any hit obtained by the assassin rogue will also be a critical hit as long as the target has not had a turn in the combat yet and is thus still surprised.
Attacks, in general, do not occur outside of initiative since attacks=combat encounter=initiative
To be slightly pedantic, "If they decide that the creature you are attacking is surprised then they do not get to take a turn on their first initiative" is incorrect.
If you're surprised, you can't move or take an action on your first turn of the combat, and you can't take a reaction until that turn ends. The distinction is that after your turn ends, you are no longer surprised and can take reactions (like casting shield spell, Battlemaster Parry ability, etc).
I have had situations where a wizard was surprised, but rolled exceptionally well for AC and was before an enemy swashbuckling rogue. The rogue moved and attacked, hoping for sneak attack since they were one-on-one, but the wizard was able to react and cast Shield. This de-railed the combat for 2 minutes while I explained this ruling.
I thought I said that? Though, I could have made it more explicit.
"When surprised, a creature can not take reactions and can not take any actions or move on their first turn. After their first turn, the creature is no longer surprised."
A creature who is no longer surprised can use reactions - which happens after their first turn in the combat has passed - though I didn't state it explicitly so thanks for the clarification. I did say that surprise ends after their first turn.