Keep in mind that domain authority is Moz's metric. It's intended to predict how well your site might perform in search results. You might be able to rank just fine without a high DA.
To answer your questions as they are numbered:
1. It's possible. Low quality links are probably low quality because they lack relevancy and are low on trust.
2. Not necessarily. A natural link profile will have all kinds of links: some good ones, some decent ones, and some not so great ones. But at the same time, I wouldn't actively pursue low quality links. I just don't panic if a scraper links to me and start tearing my hair out over my domain authority.
3. Perhaps. Avoiding websites in "bad neighborhoods" is generally recommended. At the same time, you could probably spend your hours more effectively than reviewing everyone's link profiles.
4. I wouldn't avoid directories "no matter what." I wouldn't avoid anything "no matter what." Consider what the benefits are against the risks and go from there. With any directory, consider if it looks like a resource people would actually use. If it's just there for links, don't bother with it.
Just think for yourself with these things. Don't fall in love with bumping up metrics.