The answer to the first question is: more observations.
That is, what process and techniques are used to calculate and increase the accuracy of our collision threat estimation?
You just have to know the orbit.
But the orbit of asteroids is very complicated to predict, because they are influenced, sometimes dramatically, by the planets and by smaller bodies, like other asteroids, that we might not be able to see.
On top of this we cannot measure directly the orbital speed of an asteroid. We can only observe changes in the angular position. If we know the distance (which might not be easily measurable) we can at least put a lower limit to the speed.
The basic steps to study the orbit of an asteroids are:
find it and try to get a first guess about the position and velocity (this requires multiple observations far enough in time to be able to measure an angular movement)
plug the position and velocity info (and likely their errors) into a computer simulation that simulate the solar system dynamics. Fast forward the simulation to predict the future orbit.
observe again the asteroid and compare its position with the simulation. Re-estimate position and velocity. Go to point 2.
As the orbit can change significantly a continuous interplay between 2. and 3. is necessary.
Once you have an orbit you can estimate the probability for the asteroid to pass near enough to the Moon or Earth to be deflected and hit us.
But thankfully the solar system is huge, we are tiny and moving between the much much larger Sun and Jupiter :D