...at least as far as the asteroid is concerned.
Listen close. The "anomaly", aka the asteroid in the film, is a large fragment of what was once a Pluto-sized dwarf planet. A few decades before the movie, two dwarf planets collided and shattered into pieces, with the anomaly being one of the largest chunks. This happened sometime before 1970; when it happened, either we didn't have the technology to see it (no space telescopes, etc.), or (more likely) we didn't realize what it was and why it would eventually be important.
- "And beg'n your pardon sir, but it's a big-ass sky." It really is a big-ass sky. For fun, let's assume that the asteroid hits the Earth today (early-mid 2010ish) and that we detected it when Armageddon came out (1998). That would mean that it's 7.05 × 10^13 miles away. Now Texas is 773 miles wide. And object the width of Texas, at that distance, would be .0000000000000001322% of the sky (the area of a 773 mile wide object on a 12-light-year sphere). Scaling math is left as an exercise to the reader (Surface area of a sphere is 4*PI*r^2, and (assuming a spherical object (i.e. asteroid)) the portion of that sphere it takes up is PI*r^2)
Now you've got a big chunk of molten rock (a collision between dwarf planets would melt both planets) hurtling through the solar system. As the rock slowly cooled and coalesced, friction built up. The rock didn't come entirely back together; it reformed in the shape of two VERY loosely bound chunks, and the rock was riddled with fault lines all over the place (which explains the earthquakes and geysers that the drillers-cum-astronauts had to deal with).
If you're wondering how a simple nuke drilled 800 feet into a Texas-sized rock could shatter it into two equal halves - it was two equal halves to begin with, and friction barely kept the halves together. Hell, it was already coming apart when Bruce Willis's character put the nuke in there. The energy of the nuke was the "straw that broke the camel's back". The force of the explosion was just enough to overcome the friction holding the two rocks together, blasting them just far enough apart to just dodge the Earth on both sides. Gravity will probably bring them back together, but by then the asteroid will be heading on its merry way out the other side of the solar system.
- It makes a little more sense than that if you listen to the DVD commentaries etc. The asteroid has travelled the galaxy for several million years, which excuses almost all its weird attributes. (Maybe not the grass.) It's full of gas too, and the nuke hits it at a fault line that makes it crack while all the gas go boom. I'm still wondering how they got the halves to fly in the right directions not to hit Earth, though.
- They can't. The amount of energy needed to push the halves of a rock the size of Texas that far apart when the Earth was already close enough to fill half the asteroid's sky would be more than if a nuke ten miles wide was set off, let alone a burp from a big gas pocket. All you'd get was two slightly separated impact points instead of one. Let's face it, this film just takes place in an Alternate Universe with different laws of physics. Only possible explanation.
- Yes, they fail physics forever. However, the base idea (splitting an NEO into two halves that both miss the Earth) is viable idea Warning: Do not operate heavy machinery after reading the articles linked But (allowing for  technology to get them there) nothing is overly far fetched outside of visuals. Aside from Bruce Willis somehow being a genetic ancestor of Liv Tyler.
- I don't know, look at Liv's actual father, Steven Tyler. Is that any less plausible?
- The wider galaxy may have some strict laws similar to the Federation's Prime Directive, preventing them from overtly interfering with a "developing" species. The first species sent in the asteroid as a form of plausible deniability; it was also engineered to be incredibly hostile in the event that humans somehow visited it. Meanwhile the second species was waiting on the humans so they could do the same; the two halves only miss the Earth because the aliens took advantage of the nuke to shift their paths.
- The two species are in a Cold War and the Solar System has a strategic location or resource that neither can access due to the presence of humanity. The first species was planning a land grab once humanity died, not realizing the second was onto them. Ensuring humanity's survival stops the land grab and also embarrasses the first species when their plan falls apart.
- The asteroid was hurled toward Earth by Unicron. This would explain why it's much bigger than any actual asteroid (Ceres, the biggest asteroid, has a radius of "only" 487.3 km) and why it looked like a chunk of a much bigger body: because it did not come from the asteroid belt! It was actually a chunk of a planet in another solar system, that Unicron destroyed.
- Or conversely the asteroid is/was in reality either a Decepticon version of the Ark sent to rescue Megatron before the arrival of the Autobots on earth (which if the general timeline of the current Michael Bay films took place in realtime) 6 to 7 years later. Or the asteroid was an actual 2nd giant Decepticon that happened on earth to destroy the planets inhabitants and process all of earths resources and convert our sun for energon.
- There will also be the military and a secret government agency.
- However, a poster of Bad Boys II (also directed by Michael Bay) appears hanging on the wall of Sam's dorm room in Transformers: Revenge of the Fallen completely throwing a wrench in setting up a "Michael Bay Universe Infinity War". Still would've been an interesting idea though.
The kid in question is Harry Stamper. Most, if not, all of mission's crew members are imagined adult versions of his friends or classmates including his best friends A.J. and "Chick". Harry dreams of being an astronaut, but believes he will most likely wind up in his family's oil drilling business. To cope, he envisions an apocalyptic scenario where he's The Hero and he and his friends are being called to save the world for all humankind. All of the movie's massive liberties with science are merely the kid's incredibly loose comprehension of astrophysics, astronomy, aerospace engineering, etc. Also, he's a fan of Aerosmith.
- Alternatively, the kid is actually A.J. who has a crush on The Girl Next Door, Grace Stamper. Ever since A.J.'s father passed away, Grace's somewhat overprotective, oil driller dad, Harry acted as his surrogate father figure even though he's not to crazy about him making moves on his daughter. The movie's plot is then A.J.'s convoluted action fantasy that ultimately has him wind up marrying Grace.
- Come to think of it, the kid might also actually be Michael Bay.