My impression from the show was that galactic rotation causes some mixing (true) leading to shorter transit times between solar systems than would otherwise occur. The actual time calculation depends critically on what speed the object is traveling and I have no way to estimate this. I would need to know the mass and point of origin of the objects (as well as probably a semesters worth of orbital dynamics). Back of the envelope, it would take Voyager ~4Myr to get to Alpha Centauri. As I am skeptical of interplanetary panspermia, I must admit I'm not paying close attention, but I have never heard (serious) talk of interstellar panspermia.
As to the Late Heavy Bombardment, take a look at:
Llyd's paper proposing reseeding: ICARUS. Mar2003, Vol. 162 Issue 1, p38. 9p.
Oleg and Steve's paper against sterilization: Nature. 5/21/2009, Vol. 459 Issue 7245, p419-422.