I really can't comment on primary schools. Although, I have a friend who works as a TA at Sussex Road, and says it is a wonderful school (I think a Good Ofsted score). We chose South Tonbridge because my daughter goes to Weald, and the first choice for my son is The Hayesbrook Academy.
We did look at houses in North Tonbridge (and in fact, offered on one on Hadlow Road, but the survey turned up a multitude of problems), but our personal circumstances meant South was a better fit.
We love South Tonbridge. After an initial "lay of the land", we focused our search around houses on both sides of Pembury Road. These houses had larger gardens than the ones that we saw to the north of the station. We can walk to the station, to Sainsbury's and to town so easily.
We don't hear any noise from the bypass. I will say though, as we are so close to the station, and parking is unrestricted on our street, that parking is an issue. We have a drive (for one car only), but for friends to come and visit during the day, it becomes an issue. In advance, I will park outside our house and let my friend(s) park in the drive to avoid them driving for hours (slight exaggeration!) to find a space.
The nicest road in South Tonbridge (as told to me by the EA, and not where I live!) is The Drive. But any homes in this area are lovely, I think. But I am now biased!