WMG / Star Trek: First Contact

Not all of Picard's Borg implants were removed.
He can hear the Borg because he still has receiver hardware installed in his brain that allows him not just to pick up the Collective's communication stream, but decipher and understand it well enough to glean the specific information he needs from it. Presumably, while Dr. Crusher and Starfleet Medical did the best they could, some of the Borg hardware was too deeply integrated to be safely removed.

Picard not wanting to destroy the Enterprise is more than just revenge.
He was keeping a promise he made to Kirk, not to let anything getting him off the ship.

The Crew of the Enterprise is the source of divergence into the Mirror Universe.
The Enterprise crew was present to ensure that the events of First Contact went off without a hitch. Along the way, they managed to inspire Zefram Cochrane with tales of the future and how humanity manages to get past the post-apocalyptic hell-hole they are currently in. Thus informed, he greets the Vulcans in peace. Now, imagine how Zefram would have been with no Borg attack and no Enterprise crew there: a drunk depressed asshole living in the remains of what was once a superpower nation's nuclear arsenal. When aliens descend from the sky, using a highly advanced starship with who-knows what kinds of weapons to use on the poor, insignificant humans, odds are that he would react in a more wary, more paranoid, perhaps more violent manner (perhaps by shooting said aliens with a sawed-off shotgun and looting their ship) due to fears over what will happen now that highly advanced aliens have landed in the aftermath of World War 3. Thus, the crew Set Right What Once Went Wrong and ensured that the main timeline, rather than the mirror universe, became the 'prime' reality.
  • This is presented exactly that way in the beginning of Enterprise's "In a Mirror, Darkly" two-part episode: the Vulcan ship lands, the Vulcan greets Cochrane, and then a refilmed version of the scene shows Cochrane (or his hand double) drawing a gun and shooting the Vulcan before the rest of the crowd rushes forward to loot the ship.

The Phoenix test flight would have failed were it not for the intervention of the Enterprise crew
To be painfully blunt, Cochrane was a drunk and mentally unstable. The Phoenix was cobbled together from salvaged parts and a converted nuclear missile. In the ENT episode "Dear Doctor", Archer is adamant that the technologically advanced, peaceful Valakians just don't have the necessary skills to handle warp drive technology. Yet humanity developed it in a post-World War III shanty town built around an abandoned missile silo!

Enter the crew of the Enterprise-E, who are determined to ensure that not only does the Phoenix test flight go off without a hitch, but they also want to make sure that it happens at the exact date and time required to cause first contact with the Vulcans. To this end a bunch of late-24th Century starship engineers swarm over the Phoenix and make sure that it is in perfect working order in time for the historical flight. But this could be seen as a case of a Stable Time Loop and You Already Changed the Past.

The Phoenix does work perfectly, as one would expect given the fact that it received extensive pre-launch service from engineers from 300 years in the future! One is left to wonder if, had the Borg not led the Enterprise on a chase into the past and done enough damage to the launch site to justify sending in engineering teams to work on the Phoenix, whether the actual vessel, as constructed by Cochrane and his followers, would have actually worked, or if it would have succumbed to some mechanical failure. Lily Sloan seemed to be the only person in the whole settlement, including Cochrane himself, who took the ship and the flight seriously. Cochrane appeared to be running on drunken ambivalence and constant prodding from Lily. The actual test flight appeared to have nothing even resembling the usual rigorous procedures of a space launch.

If the whole Time Travel incident had not occurred, it seems as if odds were good that the Phoenix would not have worked, Cochrane and Sloan would have died during the failed test flight, and first contact with the Vulcans would never have happened.

The Borg weren't just looking to erase the Federation from history when they selected the First Contact date.
Rather, they wanted to kill two birds with one stone: to derail the Federation, and to acquire whatever they could of a technology which, in later eras of Star Trek, would've been entirely suppressed: genetic augmentation. While Khan and his ilk were gone by 2063, records of the research that created them would've still been available for looting, along with thousands of frozen Augment embryos that wouldn't be moved to Cold Station 12 until much later. The Borg already possess cybernetic technology far beyond the Federation's, but Augmentation constitutes an alternative form of "upgrading" they could assimilate, provided they went back to an era when the Augments themselves were no longer an active threat.

The Borg weren't trying to prevent first contact at all
Rather, they were trying to provoke a reaction from the 29th century Starfleet in the hopes of assimilating one of their ships. Their exposure to Voyager might have led to the discovery of their interactions with Captain Braxton, and they saw it as an opportunity to acquire future technologies. So, they attacked the future homeworld of starfleet (which is also the current homeworld of starfleet) in a way that would not immediately be obvious as a lure. They might even have specifically given Picard the insights to destroy the cube, so they'd have an excuse to deploy their time-sphere. Unfortunately for them, the Enterprise followed the borg into the past, which meant that future starfleet didn't need to send a timeship into the past. The queen tried to contact the First Contact-era Borg to disturb the timeline in another way (causing an earlier federation-borg conflict would have significantly changed federation history, which would remove future starfleet) to provoke an intervention, but was succesfully stopped. The remaining drones from 'Regeneration' tried to do so again, but only succeeded in setting up a stable timeloop (in fact, maybe Q deliberately introduced the enterprise to the borg at that point to cause it to become a stable timeloop.)