Welcome to the Q&A Forum

Doesn't the crawl diagnostic system tell you?  If you are talking about Google webmaster tools, under Diagnostics => Crawl Errors they will show 404 errors.  They also usually show under the Linked From column the page they found the broken links on.  If they are from external sites, you can't necessarily do anything about it, although you can try to contact the webmaster of the linking site and ask him to fix the link.  If the broken links are from your own site, you fix them.

Many webmasters also use a program called Xenu.  Do a search for it.  It runs on Windows; I don't think they have a Mac version.  It will crawl your site and identify broken links and what page they are on.  It's a good idea to run a Xenu scan on your sites periodically.

1. Yes, your link back to the other site is in good faith and good for readers.  If you don't do it too much, you shouldn't get dinged for recip linking.

2. About 4 or 5 years ago I used to see sites do this, usually using the robots.txt file to exclude spidering ot their links page.  i don't know if it;'s the "best practice" but it seems robots,txt was used more often than noindex on the page.

It's a sleazy thing to do and yes, it can cause bad blood with your link partners.  I know because on more than one occasion I informed sites about that practice being used on them, and they removed their outbound links and thanked me for pointing out how they were being played for chumps.