Solution for a readonly copy of a standalone MongoDB server
Can I choose which databases/collections are replicated ?


A replica set is a group of mongodb instances that maintain the same data set, so it is necessary for the entire set of data to be replicated between nodes.  


  Is a 2-members (a primary and a secondary) replica set eligible ?


A replica set should have an odd number of voting members. If you want to maintain only two data-bearing nodes, you would want to include an arbiter in the replica set as a third voting member. 


  Can I make our production application servers "always" read-write on the primary, and our data-analysis server "always" read on the secondary ?


All writes must be directed to the Primary node of the replica set; clients are not allowed to write to a data bearing node in the 'Secondary' replica set state. 

Specifying a read preference is a good way of directing queries to different nodes of a replica set.