What site-specific configuration settings are possible, and how have they been implemented by the various communities?

This is a little broad, so the best I can offer you is a general rule of when something is probably configurable per-site. I want to offer that with a healthy disclaimer:

Just because we can configure things differently per-site doesn't mean we should or will.

The following things are common to be just a little different from site to site:

• Post processors for text (SoundCloud, MathJax, YouTube support, etc)
• SPAM protection settings and weights (higher traffic sites tend to carry more weight)
• Migration paths
• Off-topic close reasons (mods can change these)
• Small bits of text such as the site name, topic, guidance shown in a few places ¹
• Help center articles (mods can change some of these)
• Blacklists (tag, phrase, etc) coupled with warning lists that show just-in-time help

Other things are sometimes changed depending on the volume and needs of a site:

• Required registration in order to ask questions
• Different than usual rep thresholds for certain privileges
• Fine tuning of question block and rolling rate limit algorithms
• SPAM / robot honeypot and other bot-defeating knobs
• Performance tweaks

Everything else remains pretty much the same, with very few exceptions. The goal is to provide a familiar interface to a wide variety of topics and communities - the software being the middle ground.

Most settings are established based on the Area 51 proposal and outcome of the private beta period for a site, it's more what a community shows than chooses, although they definitely have say in the matter as expressed on their meta site.

This isn't something that we'd really showcase, it's just plumbing, and I'm not going to go into it more - but there's no harm in listing it, so here it is :) This is not meant to be exhaustive.

¹ _{Internationalization makes this interesting, on non-English sites, copy changes are a bit more of a big deal.}