Is it possible to mark threads as answered?
at the moment, no. You can either 'close' or 'archive' a topic. 'archive' means it's frozen, people cannot comment on it anymore. Closed means it can be re-opened.
Both cases do mean that the thread can still be seen by searchengines, which is what you would want. (You can also completely delete a topic, but that should be reserved for spam or complete snafu's)
There's some discussion going on at discourse to have a 'solved' status, but not at this moment in time. I would suggest 'archiving' is the best way to go forward for now.
hmm, what about a category "Solved" to be added later?
There appears to be a "Tagging" plugin now. https://github.com/werweisswas/discourse-plugin-tagger
that's also mentioned at the Plugin page
That would be the cleanest option, although it appears to be a little rough round the edges. The coredevs have said that soon they want to 'bless' a tagging plugin.