How is the trajectory known with the comet losing debris (as the weight must be a factor) in the perihelion around the sun before it comes close to Earth. Will the comet slow down or speed up in it perihelion around the sun?
The mass (not the weight) is a factor, in practice the mass of the comet is so negligible compared to that of the Sun's that can be disregarded in the calculations.
Bodies in orbit speed up when around the periapsis, obeying the Kepler's second law.
The non-gravitational parameters which model the forces due to the ejection of mass and rotation of the comet are in this case high A1 = +8.93 +/- 0.12 , A2 assumed 0, where A1 regards the radial component of the forces and A2 the tangential component. See MPEC 2013 W16, this circular predicts a perihelion on 2013 Nov. 28.77929 taking into account these non gravitational effects.
If you have a look at MPEC 2013 W01 16/Nov/2013 the predicted perihelion is 2013 Nov. 28.77615 not taking into account non-gravitational effects, so the net effect seems to be a slight slow down of the comet before reaching perihelion.
It was a real shame it did not make it as one body, but there are some observers who are saying there more of the comet made it than they thought waiting for more information, checked NASA and other observatories more coming out by the hour.