>>> Why wouldn't water be one of the by products of the big-bang
To address the first part of your question –
The Big Bang (which happened 13.8 billion years ago) created ~only hydrogen and helium. So, no water existed in the early Universe (a) because there were no oxygen atoms formed in the Big Bang and (b) the early Universe was too hot to form water molecules even if oxygen was present.
>>> during Earth's creation?
Whether water was present on the Earth from "day one" of its formation (~5 billion years ago, much later than the Big Bang) is still an open question, but part of why it might be unlikely is that the Earth is close enough to the Sun that it was too hot for water to stay as ice or liquid on Earth's surface (before Earth's atmosphere formed). Water that forms in comets (further away from the Sun) doesn't have that problem, as comets would be in colder parts of the early Solar System.
The Big Bang was the formation of the entire universe. About ten billion years later, our solar system, and eventually the Earth, started forming due to gravity pulling various bits of matter together. Ten billion years is a really, really long time! The two events are completely separate.
If you're interested in cosmology, there are a ton of fascinating popular science books on the topic, and research is very active and ongoing. As an introduction, I recommend Cosmos by Carl Sagan. It's a bit outdated these days, but the main points hold and it's a fascinating read from start to finish. Alternatively, if you prefer video, check out Cosmos: A Spacetime Odyssey with Neil deGrasse Tyson. It's an up-to-date retelling of Sagan's original idea.
Two generations of stars, including heavy stars that produced all the elements up to iron (including, relevant to this particular case, oxygen), and then became supernovae, which a) produced traces of heavier elements b) blew a lot of stellar material out into the universe, to become part of new star systems. So there was oxygen around by the time Earth was formed (~4.5 billion years ago, as opposed to ~14 billion for the big bang).
CRC's GED prep book used to contain this error. Most of the book was good but the "Earth Science" section looked like it had been deliberately mangled in order to discredit opponents of creationism. Or that was my theory, anyway, it was pretty bad.
> Why wouldn't water be one of the by products of the big-bang
There weren't any Oxygen atoms around until long after the big bang, and only a handful (in cosmic terms) of non-Hydrogen atoms.[1]
H2O molecules couldn't form until Oxygen had been synthesized, which occurs when a star exhausts its Hydrogen and begins fusing its Helium[2]. So water probably didn't exist until well into the the first or second generation of stars.
> One of the leading hypotheses on Earth’s formation is that it was so hot when it formed 4.6 billion years ago that any original water content should have boiled off. But, today, two thirds of the surface is covered in water, so where did it come from?
'boiling off' seems like a bit of an oversimplifiction - yes, intuitively, hot water vapor rises - when it's surrounded by cooler denser gases. So I get that if you have a cloud of various atoms and molecules orbiting a sun that is accreting together under gravity, the iron is going to end up at the core of the body that forms, and the light stuff like H2O is going to be a big fuzzy cloud round the outside; but it's not going to 'evaporate' off somewhere else, surely? If the Earth's hot, you wind up with a ball of molten iron, surrounded by a cloud of water vapor (and O2, and N2, and so on). Once the Earth cools, that vapor seems like a good source for liquid water, no need to bring it back from the Oort cloud in comet form. Unless 'boiling off' the water means that it disappears from Earth's orbit altogether... Is the idea that it gets blown out into the outer solar system by the solar wind, or just that the sun gives that cloud of water vapor so much energy that it's able to diffuse out of Earth orbit? Doesn't the same logic apply to the other lighter substances like oxygen, nitrogen, methane... so did they come back from comets too?
Temperature is just an expression of mean kinetic energy of particles. If gases (especially light gases) are hot enough, their kinetic energy can correspond to a velocity that actually exceeds the escape velocity of Earth's gravitational well. This is how hot gases can be removed completely from the Earth's atmosphere (and then solar wind & radiation take care of removing it from there). Even at today's atmospheric temperatures, helium gas (molecular weight of 4 atomic mass units, much lighter than molecular oxygen, nitrogen, etc.) can escape from Earth's atmosphere.
Yes on your edit. The origin of Earth's atmospheric nitrogen is thought to be in part due to outgassing (in other words, gas that was stored inside rocks in the early Earth and then released from inside), but is highly uncertain & debated. It's actually so poorly constrained that it may end up being my PhD thesis topic (astrophysics and origins of planetary systems). So not even the experts know, yet, although cometary delivery in cyanide ices from comets is a leading hypothesis (similar to the delivery of water ices).
Although that explanation makes intuitive sense, it seems to be based on a massive overestimation of the amount of H2O that currently exists on Earth. A thin layer of water covering two thirds of the surface is not much compared to the entire mass of the Earth.
Isn't it possible that early Earth contained many times as much H2O as it does now (like some of those Jovian satellites that are covered in hundreds of miles of ice), and what we have now is simply what's left after the majority "boiled off"? That way, there would be no need for comets and asteroids to return the lost water to Earth.
Other comments in this thread mention vast amounts of water trapped deep inside the Earth, as well as outgassing of nitrogen after the Earth has cooled down. Perhaps the water near the surface of the early Earth did "boil off" entirely, but what we have now is the result of subsequent outgassing?
Very much what I was going to suggest, as well. But I think that would mean the Earth formed in the outer solar system and migrated inward later. Not sure if there is any evidence compatible with or contradicting that.
DNA has a 'half-life' of ~500 years. Panspermania may still be a viable theory, but it is very restricted in terms of stellar distance/time in transit.
Very true, but getting this data in a vacuum and in a stellar radiation environment is tough. Who knows what the preserving effects of space are like to DNA. However, one data point is better than none.
