The cosmic microwave background does not originate with the big bang itself. It originates roughly 380,000 years after the big bang, when the temperature dropped far enough to allow electrons and protons to form atoms. When it was released, the cosmic microwave background wasn't microwave at all- the photons had higher energies. Since that time, they have been redshifted due to the expansion of the universe, and are presently in the microwave band.
The universe is opaque from 380,000 years and earlier. The galaxies that we can see only formed after that time. Before that, all that is observable is the CMB.
The universe about halfway through recombination (it was a long process but at the halfway point, it's flips to being mostly clear), much like the universe today, has a temperature. Today the temperature of the universe is about 2.7K, but at recombination it was around 4000K. This temperature corresponds to the blackbody radiation profile of the universe. Thanks to inflation, this radiation profile is almost perfectly isotropic and almost perfectly fitting to a perfect blackbody curve (He said, perfectly overusing forms of the word "perfect" in this perfect sentence).
At the time of recombination, the universe turned from an opaque mess to clear space. But what did it look like? Today, empty space looks black. If you stare out to the farthest reaches of emptiness, you'll see nothing. The nearly isotropic background radiation produces a temperature profile for a 2.7K curve. Most of the energy of that curve is way outside the range of the visible spectrum, which is why empty space appears black. At recombination, our isotropic radiation profile corresponds to 4000 K and looks something like
I made this plot using Mathematica and you can see that the visible spectrum resides in a significant portion of the emitted radiation. So that means if you were to stare out into the abyss of empty space at the time of recombination, empty space would look like one solid, non-black colour (it's actually an orange colour). It would also be excruciatingly intense. Imagine if everywhere you looked was like looking at the surface of a star (On the bright side, you wouldn't have to worry about vampires any more).
As time goes on and the universe keeps expanding, this background colour will (relatively) soon fade (red-shift) to the black we know so well today.
If you want to see the specific colour that we would perceive at recombination or at any other temperature, go to Wolfram|Alpha and type in "Wien displacement law for [T] K" except replace the [T] with a desired temperature (4000 for our case).
Think that's cool? Then think about this: the original colour of empty space (and if you're an originalist, the true colour) is not black, it's orange. Before this, empty space was hotter, but it was opaque so we can't really talk about the colour of empty space then. And before nucleosynthesis is too short of an amount of time to count. So space is originally orange, it just shifted to black because it expanded. Mind=blown
What the hay, here's the colour we'd perceive empty space looking like:
Best Answer
The first thing to say is that the recombination didn't happen at an instant of time. Before recombination most hydrogen atoms were ionised but there were a few neutral atoms. After recombination most hydrogen atoms were neutral but there were a few that were ionised. During recombination the ratio of ionised to neutral hydrogen atoms changed smoothly. I'd guess the figure of 377,000 years corresponds to a temperature of around 3740K when 50% of the hydrogen atoms are neutral, but the temperature had to fall to around 3100K to get 99% recombination.
The calculation is described in details in this document. In brief, the evolution of the early universe is described by a solution to the equations of General Relativity called the FLRW metric. Using this equation, and observations of the current universe, we can calculate the properties of the early universe and in particular we can calculate it's temperature.
The reason that temperature matters is because it's the temperature that determines whether a hydrogen gas is ionised. If you take hydrogen at room temperature it's obviously neutral, and as you heat it the collisions between hydrogen atoms get increasingly energetic until around 3100K they get energetic enough to start ionising atoms and form a plasma. So as we look back in time, and the universe gets hotter, there's a point where the temperature reaches 3100K and the hydrogen starts being ionised.
The calculation of the ionisation as a function of temperature is complicated because the early universe wasn't in equilibrium, but physicists are good at this sort of thing (not me - I have no idea how to calculate it! :-)
Re your last question, the age of the universe is 13.75 ± 0.11 billion years i.e. the error in the calculated age is 110,000,000 years. So you would have to say the time since recombination is 13,699,623,003 ± 110,000,000 years, and obvious it's silly to give the number to more than 4 digits.