Hi @sdpapet,
It may be, but as far as we know you just have 3 files like in the tutorial, what is exactly in your reads can differ depending on what has been done to your reads after sequencing. You certainly want to make sure of this. DADA2 assumes primers are removed.
With DADA2 you can trim the 5' of your reads using the trim
parameters, however, you can only use a fixed number here so if your primers are a constant length it is probably ok to remove it with DADA2, however sometimes primers can come with heterogeneity spacers and that can add variable length that DADA2's trim won't be able to deal with.
I prefer doing all my pre-DADA2 processing with q2-cutadapt.
I personally don't think think is a poor quality run, especially for the typical V4 primers on a 2x300 run, but people have different standards 
While counter-intuitive, this can actually have the reverse effect because the more poor quality sequences you keep in your reads, the more reads DADA2 drops during its quality filtering step, as was the case in your first run. You essentially want to truncate as much as your reads possible from both F/R reads without compromising the merging which with DADA2 default requires 12nts.
Your new run does again look to have lost a lot of reads and the DADA2 output is lower than I would expect, but this time the issue is coming at a different step. You can see that this time you are actually retaining most of the your reads passed the filtering step, but you have a major drop during the actual denoising step, at least in most samples. This is peculiar to me, could be related to primers not being removed or something else, hard to say. Some samples look better than others too, can you think of any biological or technical reason for this? For ex. what is the difference between sample C1-T1R6-C
C1-T1R6-A
? What type of samples are these? How was the DNA quality and amplification process?
That being said, in some of your samples a big chunk of the reads are being lost at other steps like the merging, which begs the question are you sure these are the 515F-806R V4 primers, and not V3-V4?
Can you also please share the actual DADA2 stats artifact, ending in .qzv
file, instead of the underlying data that you have unzipped and shared as a text file? That way we have access to the provenance tab and we can actually check the parameters used. It makes troubleshooting much easier.
Let us know once you hear from your sequencing facility.