There are a couple of reasons. First, RNA is considered highly unstable, not because of any fundamental chemical instability, but because RNAses are ubiquitous in nature, so separating the RNA in a sample from all residual RNAses and then keeping that RNA sample free of RNAses can be quite difficult. In contrast, DNA is a lot more biologically stable, so converting to DNA ensures the stability of the sample's information content. (Edit: As pointed out in a comment, DNAses can easily be inactivated by chelating their metal ion cofactors, while RNAses do not require metal ion cofactors and are therefore much harder to inactivate.)
Second, PCR amplification only works on DNA, so unless you can obtain enough RNA to feed directly into your sequencing protocol, you need to amplify with PCR, and therefore you must reverse-transcribe to cDNA. In theory, you could probably adapt PCR to RNA using RNA-dependent RNA polymerase instead of DNA polymerase, but RdRp is a viral protein with a relatively high error rate (to facilitate rapid viral evolution), so an RNA-based PCR protocol would introduce far more errors than the combination of reverse-transcription and DNA-based PCR.
Third, most (all?) existing sequencing protocols are designed for sequencing DNA using DNA polymerase. Again, you could probably adapt the protocol to use reverse-transcriptase or RdRp in order to sequence directly from an RNA template, but again, RdRp is too error-prone, and if you're going to use reverse-transcriptase for the sequencing, you may as well use it initially to get cDNA and then sequence that. The overall error rate will be the same, and you will have DNA for most of the time instead of having to babysit your delicate RNA.
So basically, RTase has problems, but there's no alternative for sequencing or amplifying RNA that doesn't have even worse problems.
As for your second question: RTase takes a single-stranded RNA molecule as a template and synthesizes the complementary DNA strand, resulting in a duplex of one RNA chain and one DNA chain. Typically, after this process is complete, RNase H is used to degrade the original RNA template, leaving only single-stranded cDNA. This is then replicated using normal DNA polymaerase to form double-stranded DNA, which can then be amplified by PCR or processed in any number of other ways.
I'd just add to your great answer that it could be argued that RNA IS fundamentally chemically more unstable than DNA (particularly in basic environments) in that it is a lot more prone to hydrolysis because of the nucleophilic 2´ hydroxyl groups. This hydroxyl group is also the reason that RNAses don't need metal ions for activity unlike DNAses. One of the reasons RNAses are harder to deal with because you can't just easily inactivate them via chelation with EDTA etc like you would with most (if not all) DNAses.
Looks like you're right. "The problem, however, is that mRNA is very unstable outside of a cell; therefore, scientists use special enzymes to convert it to complementary DNA (cDNA)". Source: http://www.ncbi.nlm.nih.gov/About/primer/est.html
But if (when) large-scale single-molecule sequencing becomes viable, it might be possible to sequence RNA directly?
Single-molecule methods based on non-polymerase-based reactions might one day be usable to sequence RNA directly. With today's technology, if you could obtain a sufficient amount of RNA without needing PCR, you could maybe theoretically sequence it via SOLiD's sequencing-by-ligation technology if you could adapt the protocol to be RNA-friendly. In that case you would never involve any kind of polymerase or RTase.