While we can't 'solve' the three body problem, we can make increasingly accurate estimations. Enough to usually get orbital mechanics right in the short-term.

For a setting where we were getting it right until we got it wrong, see Niven's "A World out of Time".

Only if the target was far and we did it in one shot. Otherwise you’d just measure the state of things and make corrections for your next maneuver, the way we do for trips outside Earth’s orbit today.