Now with PostgreSQL 9.4 there isn't really any reason to use MongoDB (JSONB is better suited than BSON). For any other use CouchDB.
There are a few. Mongo has excellent integration with Hadoop and is becoming very popular in the big data analytics space. Likewise it is gaining traction in the EDW space as a result of their partnerships. Also it stills remains one of the simplest databases to setup and use making it still my goto for hacks/spikes.
Eh, more complex and less secure the sqlite which if you want can store json as string values and provide full text search.