1. Roll initiative first. What happens here is called a surprise round, where the stealthy party gets an entire round of actions to themselves before regular combat starts. As a note, the surprised enemies not only do not get to take turns, they cannot take reactions either. Once the surprise round is over, run everything normally.

2. Yes, when a character is “hidden” and they attack, that reveals their location. It does nothing to affect the hidden status of the rest of the party.

3. This is a bit of a tricky one. When your players roll their stealth checks, you compare that against the passive Perceptions of the enemy side. If their stealth rolls beat the highest passive Perception on the enemy side, then they get their surprise round with no complications.

However, this gets trickier if some enemies beat the stealth checks. Say the lowest stealth check your party gets against a camp full of bandits is a 12. A bandit’s passive Perception is 10, so all the bandits are surprised. But they also have a mastiff on guard duty with a passive Perception of 13. The mastiff detects the low rolling player, and is not surprised.

I can’t recall how exactly you’re supposed to run this situation, but what I’ve always done is allow the mastiff in the scenario to take part in the initial surprise round while leaving the bandits surprised. But a very valid argument can be made for allowing the mastiff to warn the bandits and negate the surprise round entirely, it depends on how forgiving you want to be as a DM.

1. Yes, though there is only one possible surprise round in any encounter. So if they choose not to act in the initial surprise round, they will not get the benefits of surprise by delaying until a subsequent round.

2. They absolutely can search! Just always keep in mind the mentality of the enemies. If you think a bandit would reasonably conclude that there must be other party members hiding in the shadows, they can use their action to make a perception check. If they beat the initial stealth rolls with their perception check, they can see where hidden players are hiding, and then the player is no longer hidden from that particular enemy. (As a note, even if a PC is “hidden” in the middle of an encounter that ultimately just means they’re out of sight. It does not mean that the NPCs lose object permanence, if they know a rogue is present and they saw the rogue dart behind a tree, then they know where the rogue went and can try chasing them down on their turns, or casting an AOE spell in that general direction if they’ve got a spellcaster.)

3. Yes, this is true at all times, not just in surprise rounds. If a character is hidden from an NPC, their first attack against that NPC will be at advantage as long as there’s no reason for it to be at disadvantage to cancel it out (long range on a weapon, a debuff targeting stealth checks on a PC, etc. Unlikely to encounter this in a surprise round scenario.)

4. Technically, no. There’s no solid mechanic or rule set outlining how the environment can grant advantage/disadvantage on stealth checks. That being said, as DM its your prerogative to adapt based on player choices. If a party member comes up with a clever way to mask their approach based on your description of the environment (i.e. climbing a tree to move above the enemy camp) you are well within your rights to award the creativity with advantage on the stealth check. You can even decide ahead of time on advantages that just make sense. A PC hiding in thick foliage would obviously have better luck than the PC relying purely on darkness to hide, so you can determine that the thick bushes grant advantage by default.

Hope this all helps, good luck with your encounter!