I read that mutations are more likely to occur on "the strand that DNA polymerase replicates discontinuously". Does DNA polymerase always replicate the same strand discontinuously and, if so, how/why?
In bacteria or organisms with only one well defined replication origin and a circular chromosome, yes for a given DNA region the same strand is replicated discontinuously.
In high order animals, which replicate chromosomes using several origins of replication (ori), this is less clear as the way ori are recognized is still not fully understood but seems to involve DNA structure. Chromatin structure is dynamic which means ori are also likely dynamic which results in defined regions of the DNA not likely to always be replicated discontinuously from a well defined strand (i.e. DNA replication might not always start at the exact same sites).
First off, as you probably know, DNA is always transcribed in a 5' to 3' direction. The why is nicely answered in this post.
DNA replication forks start at replication origins. Replication origins are recognized by origin recognition complexes. The way DNA replication occurs is by having two DNA polymerase complexes traveling in opposite direction (i.e. two bidirectional replication forks) from the origin as shown in the following schema:
This means that given constant origins of replication in an organism (like bacteria which usually only have only a single origin of replication) and given the DNA is replicated in a 5'-3' direction, same strands will be replicated discontinuously from the origin (i.e. in the schema, the bottom one on the left and the top one on the right). A global view of the replication at the chromosome level in bacteria is shown here below:
Now for more complex organisms, the origin of replication is not a single site but rather multiple ones across chromosomes (30'000-50'000 ori in humans). Where does DNA replication starts exactly, this is still not fully understood. DNA sequence is likely to play a role but no highly conserved consensus sequences were found and it is believed that origins of replication are linked to the structure of the DNA instead. In essence the chromatin structure is what seems to dictate where DNA replication starts and as the chromatin structure is dynamic within a cell, it is likely that origins of replication are also dynamic therefore making the identification of the discontinuously replicated strands very difficult (Méchali M, 2010).
Finally if you wonder what happens when two replication forks collide, well the answer is simple, replication stops and the two DNA polymerase complexes leave the DNA.