Unless you have a very good reason to add an extension, drop it.
are resources supposed to have a "name.prefix" structure?
Not that I know of. Normally not. Resources are just a concept. A custom resource format may have that extension requirement, the other would not. It will depend.
As for SEO, the short a link is, the better. It will increase relative weight of keywords. An extension would make links longer by 4 characters or more.
Do you know of any clients that can't deal with this and start looking for /index.htm or .html?
A problem may arise if you decide to support multiple entry points.
www.exampledomain.com
www.exampledomain.com/index.html
www.exampledomain.com//index.htm
www.exampledomain.com/index
These are all different urls to search engines. Some people will be linking to you with the shortest name, the others will use the other version. Then ultimately there will be different inbound links pointing to your site start page which will essentially be the same. Search engines will detect it and see it as content duplication. Consequently, your page rank will be divided between several url versions. Finally, all except one will likely be dropped out of their index entirely. To deal with this situation, decide for one "true" url and let others perform 301 redirect (moved permanently) to the "correct" url.