Can a methylated strand of DNA be replicated without removing methylation? Does it make any difference if the strand is methylated or not (during replication)?
Absolutely. It's a pretty cool process, actually. Most (well...) DNA methylation occurs in the context of what are called CpG; that is, a C (Cytosine) followed by a G (Guanine). Because C and G are the Watson-Crick pair for each other, the sequence on the opposite strand will also be CG. Usually, both Cs are methylated, which turns out to be rather critical for maintenance of methylation.
DNA replication occurs through a semi-conservative mechanism, which means each old, original strand is copied and paired with a new strand. The new strand has no methylation on it, however; it is at this point that the enzyme DNA methyltransferase (DNMT1, specifically) comes into play. DNMT1 finds the CpGs methylated on one strand ("hemimethylated") and methylates the other strand, providing complete inheritance. Here is a nice summary, and this is a review that (briefly) summarizes the evidence that CpGs may play a role in irigins of replication (the jury's still out).