If you recall in the Star Trek "First Contact" Movie, in the opening scene, we see Picard, he is dreaming he is on the Borg Ship, he can hear all the voices.
He then opens his eyes, walks to the bathroom sink, where he splashes water on his face. We see a bubble form in his cheek, and then a Borg implant pops out of his cheek. He is freaked out! He awakens from what he believes was a dream, yet, the subspace communication that awakens him is very real. Before the Admiral can tell Picard the Borg have attacked, Picard cuts the admiral off, and quips, "Yes Admiral, the Borg."
He knew the Borg attacked without ANY information from anyone! I would argue that although the implants were removed, much of the technology laid dormant in his brain. When the Borg were within range, he could hear the voices of the Borg, and knew they were close.
We all know that the Enterprise is ordered to patrol the Neutral Zone, while the Borg attacked the Federation. Riker interrupts Picard, where he is in his quarters listening to opera very, very loud. Was this to drown out the voices in his head? Riker says, "We have completed our first survey..." Riker continues, "what are we doing out here in the Neutral Zone anyway?" Picard responds, "It would seem that Starfleet has every confidence in the Enterprise, yet they believe introducing a person who was once assimilated by the Borg into a conflict against the Borg, would represent a potential danger to the Federation." We all know what happens from there, as Worf is saved and Picard and the Enterprise save the human race.
Jumping back a few seasons to when Picard was assimilated. If you recall Picard was aboard the Enterprise, as Locutus. Where we see Picard raise his artificial Borg arm, which was not amputated, but added to the end of his hand. Riker and Worf step back, and Locutus says, "There is no need for apprehension as I intend to cause no harm to anyone." He then pauses, looks right into the camera and says a second time, "No harm," clearly indicating that Picard was fighting against the Borg commands.
Crusher discusses returning Locutus back to Picard. "It would be just a matter of microsurgery, and I could do it, however, with the implants active, it is impossible." This suggests that once he is removed from the Borg Collective, all of his physical implant" were removed, yet the implants were operating on a potentially different level (perhaps a fourth dimension)?"
Crusher and Data summarize the physical connection between the Borg and Locutus operates very similarly to that of how the Enterprise uses its transporters. If that theory holds true, then the Borg can add or subtract information and components as easily as O'Brien can deactivate a phaser that is armed and potentially discharged, and render it inactive.
The Borg Queen is seen at the end of the "First Contact" movie speaking to Picard. Picard seems stunned as he tells her, "but all the Borg and that ship were destroyed?" The Borg Queen responds, "How 3 dimensional you are to think the Borg only operate on the same quantitative (three) dimensions as humans," and how she expected so much more from Picard.
Picard then offers himself, without resistance to take his side next to her as her equal. However, The Borg Queen responds, "I have found an equal in Data, as he understands the nature of the Borg." Data, who is deceiving her (with the exception of the 0.80 milliseconds whereby he was tempted to join her), commenting on how Picard "will make an excellent drone."
I apologize if I am jumping all over the place, and or confusing anyone. I will be happy to clarify anything that I wrote to anyone who desires it?
In answer to the two question's asked regarding Picard and the Borg.
As to Question 1; did the Borg remove any of Picard's body?
That question is nonsensical as clearly some part of Picard's brain had to be removed in order for the implants to take the space in his head. Since we only use roughly 15% of our brain, I would guess that the parts removed were not essential to Picard's functionality, but most likely were closely tied to aspects of the brain's ability to control and dominate Picard. As Picard could not stop Locutus from destroying half of Starfleet at Wolf 359 and the Borg were able to download every thought, tactic, strategy, and knowledge which Picard possessed, as if his brain was a giant computer. Yes, I believe the Borg did remove some of Picard's body parts, most primarily the space needed to put effectively rewrite Picard's DNA, and force Picard to commit unspeakable atrocity's against Starfleet (which we never got to see). I believe we didn't see this because it would have created resentment amongst Picard fans. That even though he wasn't in control, it was his knowledge and skills that made the Borg so lethal against Starfleet.
I don't believe any other body parts were removed, as with his arm, they put the implant at the end of his fist, rather than amputating his arm (which would have provided a direct connection to his right extremity).
Question 2., Was Picard's body left with any implants? Most definitely! I believe there were dormant area's where Borg implants remained. Particularly the area's connected to where Picard's DNA was being overwritten.
As it pertains to Picard's ability to have children. I believe that episode was flawed. As in one of my favorite episodes, Episode 25, Season 5, "The Inner Light" Picard lives a complete life, spanning in excess of 50 to 60 years. Whereby he has a family, children, and grandchildren. And Picard's own statement, upon regaining consciousness, "I never believed I needed a family to complete my life, yet after this experience, I cannot imagine living a life without them."
Therefore, once Picard completes his career as captain, and or moves on to being an Admiral (where he isn't flying all over the galaxy). I believe a family would be something of a priority for him.