I much prefer how 3E handled surprise. 5E surprise makes little sense. I really hope Pathfinder puts out a new streamlined edition of their game at some point. Some of things in 5E are annoying such as their version of surprise not really being surprise unless you win initiative even though they don't know you're there...except somehow they do know you're there before you attack because they won initiative. Yet you get advantage for using stealth because Unseen Attacker, but not in the surprise round because they know you're there and shouldn't give you advantage. And the ridiculousness of this ruling and how it interacts with Stealth just continues on. This is definitely one of those things Mearls and company did not imagine very well and it disrupts a key class ability that is hard to use in a party environment to begin with.
The fiction this ruling encourages just makes my head hurt.
1. Roll Stealth.
2. Target fails to beat your stealth. He does not know you're there.
3. You attack. Surprise occurs.
4. You and target roll initiative. Target beats on you initiative.
5. Target takes his turn doing nothing but talking about the unseen person that surprised him that he still can't see, but he got to take a turn because that is what the RAW says.
6. Assassin shoots with advantage because Unseen Attacker from using Stealth. Or does he get this? I don't know. The target won initiative, so he must know you're there. But the RAW says unseen attacker from Stealthing. But he won initiative.
7. Target uses reaction to cast a spell if he's a caster, but do nothing if he isn't. Gets hit with advantage for sneak attack damage, but not crit. Sneak attack because rogue gained advantage because target was unaware of his location, but somehow avoided Assassinate because he already acted allowing him to avoid the worst of the blow, even though he didn't know he was there.
It's just ridiculous. It's like having your cake and eating it too. You get to know the Assassin was there if you win initiative, but the Assassin still gets advantage form you not knowing he is there, but you still get to act after your turn as though you know he is there. It's making my head hurt.
Not quite.
You certainly don't learn of a hidden character's location just because your turn has come up. Until that character has done something to reveal his or her position, or you have taken an action to search for that character (which you can't do on your turn if you are surprised), the character remains hidden. So, in your scenario above, as you outline it, the assassin would still have advantage, because the assassin is hidden. So sneak attack, but no auto-critical hit. That's all very clear in the RAW.
Now, as to the specific case of:
1. Assassin is perfectly hidden
2. Target is unaware
3. Allies of assassin are either far away or also perfectly hidden, waiting for the assassin to take a shot
4. Assassin fires at the target
Here, one of the core rules of 5e,
common sense has to come into play. When everybody agrees that something should happen a certain way, that is how it should happen. Those are, in effect, the rules. In the situation above, the question is all about
when does combat start.
I would say, combat starts, in this case, when the arrow hits the target. Until that point, there have been no hostile actions. So there is no initiative roll, because, until that arrow punched through the back of his neck, that target hasn't had any reason to do anything.
Now, once the target gets hit (and is surprised, so advantage, sneak attack, auto-crit), combat begins. Everybody rolls initiative. The target isn't surprised anymore (if he's still alive), because the attacking assassin has revealed him or herself
before the upcoming combat round. Combat proceeds as normal, and the allies of the assassin can now take actions, in initiative order, along with everybody else.
Remember that, in general, a combat turn includes not just an attack impact, but some degree of movement leading up to that impact. On most cases that surprise comes up at my table, a lot of things happen during that first round of combat other than actual weapon impacts. Characters charge out of hiding. Swords are raised above the head (in order to be lowered onto unhappy victims). Information is exchanged. All of this can happen, but, if the targets were unaware
before the start of combat, they can still be surprised. Usually, the party would sneak up to within 25 feet of their targets (or let their targets get within 25 feet of them), and then CHARGE! At this point, the targets know the party is there. But, they're still surprised, so they aren't able to take full actions. They're caught off guard. However, if they are sufficiently quick on their feet (luck and natural initiative bonus), they might be able to at least
react before their flat-footed ambushers can act fully. Hence the initiative roll. (And yeah, the potential nerfing of a 3rd level power for the assassin that can be used
every combat without any need to short rest, that yields 6d6 of damage, and scales as the assassin levels up.)
Now, all of that feels very complicated as I write it out, and maybe it is. The point is that I started with two things. The rules and common sense. I applied the rules
and common sense to situations (at my table), and the above is what I came up with. You might come up with something different. So my ruling, and therefore my interpretation of the rules, is based on my base rule, whatever happens at the table has to make sense. The rules aren't there to override that. They are there to support that and provide consistency to that.