1. is probably because you're using a Firefox feature. We don't have a find mode (well, we do, but it's terrible so I left it unbound). Ctrl-G/Ctrl-Shift-G cycle results in the Firefox feature. There are some function keys you can press too but I can't remember them.
2. Yeah. The annoying thing is it used to but apparently that was an oversight and there is some security hole there that Mozilla fixed. Amusingly, they have a secret god-mode permission that they give to their own Screenshot extension that makes it work on PDFs. A chap from OneNote was complaining about lack of PDF access the other day so maybe Mozilla will find some sort of solution.
We could bundle pdf.js with Tridactyl, hijack requests to PDFs and send you to our own reader, but I haven't been annoyed enough to resort to that as I personally just use Okular which is far better at rendering anyway.
Happy to chat in this thread or on Riot.