How do I search a particular Board or Thread for messages posted within specific date range ?

See this discussion for how to search for messages within a time range:

http://lithosphere.lithium.com/t5/developers-discussion/quot-Realtime-quot-metric-amp-for-a-certain-period-via-REST-API/m-p/53904/highlight/true#M1779

That approach should work with both the CommunitySearchContext and NodeSearchContext "messages" methods. So you should be able to search the entire community, or drill down into a specific category or board/blog/idea exchange/etc.