https://naif.jpl.nasa.gov/pub/naif/toolkit_docs/
IEEE double precision (64 bits) gives 15 to 17 decimal digits. (Note, by the way, that the numerical integration used to produce ephemerides uses at least quadruple precision, but that’s not really about trig functions.) As for how many are “needed”, well, that’s an excellent question, especially since not everything is measurable to such a degree. Like, if I type in “mass of the earth” into wikipedia, it says (5.9722 ± 0.0006)×1024 kg.
ETA How about this: if you consider the current International Celestial Reference Frame, we are talking about less than a milli-arc-second, so at least 10 digits of trigonometric precision are absolutely required.
The good news is, you don’t need the mass of the Earth (or of any other celestial object). What you need is the product of their mass and G, which is known to much, much greater precision. The uncertainty in the mass of any celestial object is almost entirely due to uncertainty in the value of G.
And how much precision you need for space flight depends on how much course correction you’re willing to do. Start off by aiming in the direction and with the amount of thrust that your calculations, to whatever precision, allow for. Continually monitor your position and velocity, to as much precision as you can. As soon as you’re off course enough to be able to detect it with the precision of your instruments, make a correction burn.
All that said, fuel and reaction mass for course corrections is expensive, and this thing is going to need some yet once it gets where it’s going (in fact, it’ll never get to the point of not needing course corrections, because L2 is unstable: Once it eventually runs out of fuel, that’s the end of the mission). So you want to be precise enough to use as little as possible.
Okay, so I just read the Wiki page to learn what Lagrange points are. So, an immediate question: Why put this at unstable L2 rather than the stable L4 or L5 points? Is it because L4 and L5 are likely to have (or known to have?) an accumulation of miscellaneous stuff piled up there, whereas L2 would not?
The wiki page also noted prior and future planned spacecraft at L2. Are some of these priors still there? Is Webb going to have to do some careful formation flying to avoid those?
ETA: Maybe, by the time Webb runs out of fuel, we will have the ability to send an unmanned re-fueling expedition there?
You are right. That was a bad example; the parameter for the Earth is given to 10 digits if you click on the IAU pages.
I thought the whole idea was to screen off the Sun, Earth, and Moon with a shield. An L2 location makes that possible.
The good news is that the company that won the prime role (to integrate the JWST), Northrop (née TRW), had never had a failure once final orbit was achieved. One of the best at deployables in space, more expensive than other primes, but driven by a culture of excellence.
In case there’s any confusion, the JWST is traveling to the Sun-Earth L2 point, not the Earth-Moon L2. As previously noted this keeps the Sun and the Earth-Moon system in the same direction that the sunshield can block. This is just about the furthest point from Earth something can be without drifting away into an independent orbit around the Sun.
The trajectory is planned like a high “pop fly”, where it’s reach apogee just at the L2 point. Think of trying to launch a tennis ball so it will be at zero velocity just as it reaches the observation deck of the Empire State Building.
If the Ariane 5 had failed in any way, there would have been gnashing of teeth and rending of garments. At least that hurdle is out of the way.
Space is big. Even if you left things to their own devices, a collision would be unlikely. And when you’re actively maneuvering anyway, it’s really, really easy to be absolutely sure to avoid any collision.
Probably possible with current tech. The question, as always, is just whether it’s worth the cost.
Before that point comes, though, we’ll probably come up with some new ideas for even better space observatories. And those new ideas will be guided, at least in part, by what we learn from Webb, just like Webb was designed based on what we learned from Hubble.
The main reason JWST was so expensive is because of the folding origami trick needed to pack that thing inside an Ariane 5 payload fairing.
Assuming Starship is eventually successful, it has a 9-meter diameter payload bay. A successor to JWST could be built like Hubble with a single solid glass mirror. Also, because launch costs are way lower, the telescope could be built more cheaply and wouldn’t need as much testing. I believe there has also been a proposal for a version of Starship that flares out to a 13 meter payload fairing, for an even larger scope.
I believe a Starship that is fully fueled in LEO could make it to L2 and back, opening the door for new telescopes that could be serviced on-station.
Here is a 30-minute video I’m watching that discusses the “Insane Engineering” of the JWST. At about 27:30 to about 28:15, the narrator says
I was massively disappointing to find out it will be 6 months before we get an image. It was like getting a sled for a birthday in July.
Nitpick: both the launch base and the launcher are European cooperations, with admittedly the French bearing most of the costs. The Ariane V launcher is built in and by most of the ESA member states.
Of course, it is European if it fails and French if it succeeds (according to the French). 
That’s the video I was going to recommend. It addresses an active cooling system I had no idea about as well as the multilayered shield array.
The first and most important mid-course correction burn occurred yesterday about 12 hours after launch. I haven’t seen any status reports on it but I presume it was successful. High-gain antenna deployment will happen today.
As the earth’s gravity continues to tug on it, the speed of the JWST is now down to less than 1 mile/sec. It’s now 166,000 miles from earth (about 18.5% of the total distance) but eventually its progress is going to start looking painfully slow! It will have reached the distance to the moon in about another day and a half.
With this sort of project no news is good news - if there was a failure the mainstream media would be reporting on it.
I would think that distance would be a big part of that. L4 and L5 are, by definition, the same distance from Earth as the Earth is from the sun, ie ~91 million miles. This would not only have a longer transit, but harder communication issues as well.
There probably is a bit more debris in those locations, but I don’t know if that would be enough to make it a consideration.
Not where I live: here success is spelled Erfolg and failure is spelled échec. The USA has to few languages to assign blame and claim success in such an elegant way.
To continue my linguistic/presentation/communication nitpicks (somebody has to do it, so I may well be the party spoiler) I watched the press conference on the NASA live webstream yesterday and must say it was disappointing. A young and attractive female journalist who started by pointing out that Covid protocol was being respected introduced the five or six men, all middle aged, who took off their masks when speaking into a hand-held microphone which they passed on to the next speaker, who took down his mask too before putting the micro right under his nose. 
And the French participants spoke acceptable English, but used technical terms in French. They should have spoken French with professional (!) interpretation. That is a lost battle, I know, but I must point it out.
Apart from that nitpick I am really glad that things are going well thus far. May the good fortune keep smiling to the well prepared and well rehearsed!
If you look at where the earth-sun Lagrange points are located, L2 is the only one that puts both earth and sun behind the JWST in a straight line. That’s the main consideration. The fact that it (along with L1) is also the closest such point to earth is a bonus.
At that distance, would Earth really cause any sort of interference?
It’d still be behind the sunshield in any case, or at least 30 degrees below the “horizon” of it.
I’d say that the distance is more than just a bonus. 1 million vs 90 million is not a small difference. Please note that your graphic is not anywhere close to being to scale.
Then there is the fact that things don’t just sit at the L4 or L5 points, they actually orbit around those points. I don’t know exactly how the orbital mechanics works out there, but I would think that that would mean that the telescope itself would be wanting to rotate along with the orbit, making it harder to keep things lined up.
The earth is a pretty significant source of IR emissions from the atmosphere, clouds, and the surface. This is how the planet stabilizes its temperature, and the slight reduction in those emissions by GHG absorption is why we have global warming. Since the JWST is a super-sensitive IR detector, you want it to be shielded as much as possible from both earth and sun.