Hi Ashley,
I do quite a bit of sequencing with ITS2 fungal primers on 2x250 and 2x300 kits and I also use geneious a bit for some other stuff. I use a different primer set, but the idea is similar. According to Nick's paper, your primers will amplify products for ascomycetes (183.6 +/- 46.8) and basidiomycetes (219.8 +/- 56.9). If you are only interested in ascomycetes, read joining is fine, but you will bias your results against basidios or anything else that produces a longer ITS1 by joining and discarding unjoined reads. For this reason, I would advise against joining reads for this project if you are interested in anything besides ascomycetes. I have found my ITS2 data to behave similarly, and taxa plots of joined versus unjoined data confirm the bias.
Since you are using the Caporaso design, I will assume you used your locus specific primers (plus some extra sequence to crank up the Tm) for sequencing and that your trimming is specific to 3' primer contamination. Unless doing 515-806 on a 2x150 run, I think primer trimming should be standard. I'm not familiar with cutadapt, but it sounds as though it did something at least. I typically use fastq-mcf from ea-utils. Do you use the joined paired reads script much with 16S data? I find that 2x150 reads join readily while 2x250 or 2x300 reads simply do not. I think the reason is that the quality for most Illumina runs seems to start tanking around cycle 500, but with a marked decrease in quality after cycle 350 or so. This means that read 2 for longer reads is never so good. You can sometimes compensate for this in joined paired reads (fastq-join mode) by allowing a decent mismatch (30% seems to be OK, despite Erik Aronesty's suggestion not to exceed 10%). Fastq-join seems to do a fine job of finding the best match.
One other thing I have found is that a surprising number of read pairs simply do not match for 2x250/2x300 runs. I haven't explored genome sequencing runs enough to be certain it is happening there, but I have some assemblies that suggest that quite a bit of discordance is present. This is likely due to mixed clustering during sequencing where two neighboring clusters are close enough together that signal can "bleed" into the other, and if one is producing more signal, it may overwhelm the signal from the target cluster. I found that phix is finding its way into my data this way and now take measures to remove it before I start analysis (
http://enggen-nau.blogspot.com/2015/01/bash-scripts-for-qiime-work-akutils.html). This is another argument against use of read 2. I have checked some dual-indexed data and it has far lower rates of phix infiltration which suggests it will also suffer much less from barcode bleed. I think our lab will move to a dual indexed design very soon. I confirmed this in some of my own data by blasting read pairs and finding they came back as different results. I note both of the reads you supply that don't align come back as probably fusarium something, but if you try to assemble the reads by eye you can see they do not match (try to match up the strings of polyA/T). I'd guess your samples come from a soil somewhere (possibly agricultural) where fusarium might be common. Conversely, if you look at the pair that do match, it is easy to see that the tail of the second read aligns perfectly starting at the third line of read 1. Conclusion, the pair that is not joining simply does not match!
Geneious assembler is iterative and a little "generous" in my opinion. For genome assemblies, I have seen it leave in adapter contamination which it then uses to scaffold together a contig that would otherwise not have assembled. I like it for genome work due to the automated kmer optimization, but I wouldn't trust it with my amplicons.
Hopefully this helps.