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.