>>920
If you go to network -> data -> cookies you can create a pixiv cookie and then copy your login info from your browser(F12 -> application -> cookies on chrome). I think all you need is the PHPSESSID and __cfduid. I'm not sure if there's a way to log in through the companion.
>>922
A parser is what looks at the code in a page and takes stuff out like links and text.
Take this page for example https://gelbooru.com/index.php?page=post&s=view&id=5825934
If you press f12 you can see the html code that the page is build from. If you download that page through hydrus the parser will look at the html it receives from that page to detect the link for the image it should download and the text it should interpret as tags for that image.
>>925
The PTR is just a centralized repo for the tags. The PTR tags themselves are saved on your computer.
It is a single point of failure, but due to the offline nature of the data it isn't that bad.
>>928
There's file relationships like alternates and custom ones, but I don't think there's a good way to have before/after relationships that way.
There's the page: and title: namespaces which are sometimes used in the PTR though. Those are pretty useful for image sets and mangos.