You're in luck, because there are only 2 basic things said about when someone is surprised, and both are in your favor.
The first one is:
The GM determines who might be surprised.
So yeah, you explicitly get to decide in the end. If you don't think it's reasonable for the enemy to be surprised, they aren't.
The other one is:
If neither side tries to be stealthy, they automatically notice each other.
And your party does not seem to have any interest in being stealthy, so they don't get to surprise anyone.
There's one last part, that's also interesting and probably key to explaining why they don't get surprise, the next time it comes up. (Emphasis mine)
Any character or monster that doesn't notice a threat is surprised at the start of the encounter.
The question is not "Do they see an attack coming?" but "Did they notice a threat?". Meeting an unfamiliar person, in armor, while looting a dungeon is definitely noticing a threat, whether you see their attack or not.
Now if your player wants to stab the bandit in the face before he has time to act, that's what a high initiative roll is for, not a surprise round.
It stacks
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
If you take the attack action, you attack twice.
At the start of your first turn of each combat, you walking speed increases by 10 feet, which lasts until the end of that turn. If you take the Attack action on that turn, you can make one additional weapon attack as part of that action. If that attack hits, the target takes an extra 1d8 damage of that weapon's damage type.
If you take the attack action, you can make one extra attack.
Nothing in these two statements prevents the other. One changes your number of attacks to a flat number (2), the other says "whatever it was before, +1".
2+1 = 3, you get three attacks in the first round of combat, and one of those 3 attacks deals extra damage. You get only 2 attacks for the rest of the combat.
On multiattack:
Multiattack is a monster specific action. It is NOT the 'Attack action', it's a specific action some monsters have, that determines what it can do with those attacks. This is to prevent, for example, a bandit captain from using its multiattack to make three grapple attempts in a single round. Both your features require the 'Attack action'.
Best Answer
There are a few potential concerns with this plan. Some of them work out in your favor, others do not.
Bonus action between attacks
You can take bonus actions between attacks, per the answer to this question.
Using both hunter's mark and zephyr strike
You can apply one spell to the first attack, and the second spell to the next attack. Of course, hunter's mark will end, but you'll still have access to the additional damage and advantage from zephyr strike.
Casting hunter's mark without initiating combat
This is the tricky part. "Initiating combat" isn't quite the right phrase here; there's no action you can take that will definitively do so. Whether combat has started is the DM's decision. However, it is possible that they will decide that combat has begun once you cast the spell. This is because the spell has verbal components, and casting it may make some noise.
If you are relying on being hidden to keep you out of combat, casting this spell may give away your position, potentially costing you the element of surprise. However, at a range of 90 ft, it's entirely possible that you can do so without attracting attention.
Benefiting from Dread Ambusher
Exactly when you benefit from Dread Ambusher is a bit tricky in these situations, but you should gain its additional attack at some point. If your targets have particularly good hearing (perhaps they've tamed a creature with the Keen Hearing trait,) your targets might be the ones initiating combat. In this case, you'll still have all the benefits you describe, it's just that your opponents might have already begun moving against you before you use them.
If you remain hidden after casting your first spell, all should go as you describe.