Well, either she was unable to remove all of them, she didn't locate some of them, or the Borg collective communicates in ways that go beyond simply the implants.
She couldn't have removed them all, because nanoprobes are microscopic, and thus part of his circulatory system. It's probably like having antibodies for chickenpox after the fact.
That it was a temporary weakness due to the battle was always obvious to me (and almost all other posters, apparently) - indeed, it's quite clear, considering the situation and wording. It's contraindicated to waste lines to point out the obvious.
She couldn't remove them all actually because that would have been too inconvenient for the movie script. After all....they couldn't just have Picard be the most respected captain of the fleet with the best ship at his disposal to boot. No, they had to have him suddenly develop some superhuman capabilities as well that allowed him to telepathically "feel" what the Borg was up to. Hmmmm. Funny how none of those abilities showed up in post Best of Both Worlds conflicts with the Borg on the TNG show. Then again the Borg Queen never made an appearance either during the whole TNG run. Don't think about this too hard, folks. It never made much sense.
Picard didn't encounter the Collective between BOBW and FC, in "I, Borg" they were too far from the Borg ship that came for Hugh and in Descent they weren't Collective Borg. There was no opportunity where he'd have been in the position to "hear" the Borg like he did in FC, so there's no plot hole.
Oh, nonsense. He felt the Collective from a billion miles away in First Contact when he was far away from the battle. What's the difference? You're suggesting that a billion light years is close enough but a zillion light years is too far away? First Contact is full of ridiculously convenient touches like this. Such as Data being able to absorb machine gunfire like he was Iron Man. No damage at all to him either.
You're assuming the bullets went into Data, and weren't just smashed against his external frame and couldn't get through. We don't see him with his shirt off so we don't know if the bullets even penetrated. As for the start, that wasn't even a clear message. It was just a nightmare (it wasn't a flashback, because Picard never took a needle to the eye in BOBW).