RNA seq and using of Poly(A) or non-Poly(A) based amplification of RNA
Yes, it was odd that they said that. The second protocol with the polyA based method had another purpose though, which was "examine the reproducibility of period-regulated genes in an independent genetic background". I would suspect this was its main purpose, and not so much to look at fragment bias, which has already been done in other papers. They didnt even publish this data in the main paper, they put it in the supplement. 
Yes, this would remove any transcript without a polyA tail. 
One way of purifying is by using oligos that are antisense specifically to rRNA. The oligos will be attached to some solid phase, on a column or a bead or something, and the rRNA will stick to it and everything that doesnt is what will remain in your sample
It is not safe to say that you lose information with polyA based kits. The polyA based technique had reads that were single end and only 75bp long. This is compared to the other protocol which was 100bp paired. Reads that are 75bp will align much more easily than 200 bp (100 x 2 since its paired). However, there will not be as much certainty in the alignments, as there is a greater chance it will align to multiple locations in the genome, or align to the wrong location. So yes they have more reads aligned but I would be hesitant to make any conclusions from this. And if you look at total numbers of reads, the non-polyA method aligned more. Again, I dont think the main purpose was to show that one method is better than another.


Basically the method you choose depends on your experiment and your samples. Random hexamers will give you non-coding RNA which is important in some experiments, and that is its main advantage. It will also give more uniform coverage across the gene (no 3' bias). Poly(A) selection will give you only mRNA, however you will miss any transcripts that are degraded or partially degraded. If you have a high quality sample or a large amount of starting material this is less of a concern. As mentioned before you end up missing parts of the 5' end on some transcripts with this method. If you have a low abundance transcript that you are trying to find, polyA selection is probably best, as it is more likely to be captured and you will capture the whole thing, rather than just getting bits and pieces of it with random hexamers. They both have advantages and disadvantages. Hopefully this helps you or someone make a decision.