"...whether or not Neo would have still broken the vase..."
That's impossible to answer. The answer any individual comes to will be based solely on personal perspective, belief, religious programming, whatever. Do you believe in choice? Do you believe in fate?
"...would Trinity still have fallen for Neo..."
Only infants can be told who to love. Let's say Trinity hated a particular One candidate and he started to manifest above normal abilities. We all know right from the beginning Morpheus believed 100% in the candidate. So it's the same situation as with Neo, but she hates the candidate. Would she follow the Oracles orders, despite her feelings, and assuming the Oracle doesn't feed us poison?
"...why would the Machines have the Agents pursue him to the point of death?"
Because they had to fool us all. Remember that the Machines lied to everyone about the function of the One. They claimed the One was a messiah. They said he'd save everyone, blah blah. It was all a lie to keep Humanity from actively resisting Machine dominance. If the Machines didn't try to kill Neo everyone would have known right off something was wrong. We would have abandoned him because it'd become obvious he was a tool to further Machine dominance.
If even one shred of evidence had appeared before Neo himself found out the truth, than he would be a nobody. As it is only the fanatical believer, Morpheus, and the blinded by love, Trinity, truly stuck by his side. Who do we pray thanks to that they were enough?
"If Trinity hadn't kissed Neo..."
You've got me there. I lean toward no, but it's hard to explain how Neo could have been aware of anything going on inside the ship for such an event to be relevant. It does "feel" like it was meant to be viewed as the cause of his rebirth...
"...would the Agents have just screwed themselves..."
I don't think so. We know they were aware Neo was anomalous. They could probably literally see that fact. They wouldn't need to know the true function of the One to perform their function, which was seek and destroy. Agents are just Sentinels with a Human RSI. Broadly charged with destruction of Redpill and Exile activity within the Matrix. As long as there is a 1% rejection rate and Programs choose to run from the problems within their society Agents will be around.
"...are the Agents there to filter out the One..."
Agents are there for general suppression, this makes them the "front line" in the virtual side of the war. As a side effect they are the first openly hostile Programs working for The System to encounter the One.
"Does the Neb Hvcft have gun turrets..."
Yes.
"...so, why aren't they used..."
Because everyone still alive was busy. It was a sneak attack. Waiting for Neo to get out and firing the EMP would kill all the Sentinels at once, save a lot of ammo, and allow Neo to live.
"Crew members alive and onboard: Tank, Trin & Morph. Tank was Operating. Trinity was kissing. And Morpheus was... watching?"
Morpheus was at that time not just watching. He was standing with his hand on the EMP trigger.
Trinity was the one just standing there(until Neo died) with nothing to do..
At any rate they didn't have any gunners left, because Cypher killed the two gunners. Remember?
"Not like this...not like this..."
I just answered all your questions. Woo.