Question: Why did it take Starfleet 100 years to find the remains of the Borg and their sphere in the Arctic?
Answer: That's just how long it took. We are still digging up artifacts that are thousands of years old. It takes as long as it takes. Add to that the fact that, at the time the Borg came, Earth was still recovering from a massive war, and it's safe to assume that it took a while for complete exploration and research missions to get going full bore again. Also, they didn't know to look there. There would be no reason to simply head off to some random site in the Arctic and start digging.
Question: Throughout various episodes, the augments have shown themselves to be very strong, yet whenever they face each other they easily succumb to whatever physical violence is directed towards them. Why is this?
Answer: They're stronger than humans, but they're not invulnerable.
Question: During the opening title sequence, one of the future shots, there is a shot of a starship flying over the camera. The engines (3 large, with 2 sets of 2 small ones in between, glowing blue) and the ship's movement look very similar to the star destroyer from Star Wars New Hope's opening shot. The theme for the Enterprise's opening is pioneering and exploration, and Star Wars was a revolution in special effects (and ILM has done some shots for Star Trek). Is this a tribute or just coincidence?
Answer: It is likely just coincidence. The ship in question is designed to show the state of space travel about halfway between the flight of the Phoenix and the launch of Enterprise. It is flying over the moon, to highlight the fact that in the early 22nd century, mankind was just mastering the colonization of our own solar system.
Question: Are lithium crystal ever mentioned in Star Trek Enterprise? I stopped watching the show after the first season but am curious if this was mentioned. I know that the first few episodes of TOS mentioned the use of "lithium crystals" instead of dilithium crystals and am just curious if that was explained in Enterprise.
Answer: No, they are referred to as lithium crystals in one episode of TOS and dilithium after that. The name was changed as producers wanted something that wasn't found on Earth.
Question: Does anyone know how old Zefram Cochrane is or when he was born? He seemed pretty old in Star Trek: First Contact which takes place in 2063, and "Broken Bow" shows him still alive about 60 years later dedicating the Warp 5 complex with Captain Archer's father Henry.
Answer: Acording to the Star Trek Encyclopedia Cochrane was born in 2030. His warp flight was in 2063, and he himself disappeared from Alpha Centauri in the year 2117 at 87. Kirk met him on a planetoid in the Gsmms Canaris region in the year 2267 making him 237. The first episode of Enterprise is in 2151, so Cochrane was missing for 34 years by that time and would have been 121 at that time.