Usually it is explained, that paired-end sequencing is useful for determining with more certainty where the reads come from (which can be useful if we have many repeat regions).
However, TopHat2
For paired-end reads, TopHat2 processes the two reads separately through the same mapping stages described above. In the final stage, the independently aligned reads are analyzed together to produce paired alignments, taking into consideration additional factors including fragment length and orientation.
1) If the reads are aligned separately, doesn't that contradict the basic idea of paired-end sequencing (to get a more precise mapping) ??
2) Perhaps what I lack to know the answer is an understanding of the final stage. It'd be great if someone could refer me to an explanation. I didn't find one in the forums.
Paied-end sequencing is sampling two ends of a fragment. If you have a reference sequence available then mapping those reads on to that reference gives you a precise idea of the length of that fragment. If one (or both) ends are multi-mapping then there is no way to precisely figure out where that fragment came from. If your fragment happens to capture a splice site or a breakpoint then this can also give you useful information about those events.
There is an implied assumption that fragments going into the libraries are of a certain length. This assumption is used by some of the aligners to decide the orientation as well as "properly" mated length of those two reads.
If your paired-end reads overlap then you could simply merge them and use that single representation. Some aligners may have trouble if your reads completely overlap and extend into adapter sequences (short inserts).