Hi,
The read quality looks good. It's not common that the reverse reads are much shorter than the forward reads. Do you know what happened during the sequencing? Did the sequencing stop earlier than it should have been?
Based on your quality plot, if you trim your primers and truncate the last 10 nt of both reads you should end up with 430-440 nt after merging forward and reverse reads. If the expected amplicon size of your study + 20 nt (the recommended minimum overlapping length) is less than 430-440, you should be fine. If not, you can discard the reverse reads and use the forward reads only at the cost of lower taxonomic resolution.
No. You just need to trim the last few nucleotides, say 10 nt.
You can ask your sequencing service provider for the primer sequences. To check if the primer sequence is present in your reads, which I suspect is, you can check this forum post: