i think the pgces are forward but not backward compatible - so a primary school teacher can teach secondary but a secondary cannot teach primary. however, it's not often that you meet a primary school teacher in secondary school, but it does happen. in the same way, a secondary school pgce allows the teacher to teach post-16, but a post-16 pgce wouldn't allow you to teach in a secondary school. i could be wrong, but this is how it was explained to my partner and i when we graduated with pgces, but this was 4-5 years ago.
qts is awarded after a years further induction during your first jobs (your induction year), as it says on everybodies qts certificate when they first graduate (in small print under the big letters saying "qualified teacher status".