I am a Missouri resident and my spouse works and lives in Washington DC. Can we file jointly? If so, do we need to go for paper based filing or online itself?

For Missouri, you must use the same filing status as on the federal return, but you are able to indicate the non-Missouri income.  

For DC, you can filing separately using the federal joint return.

For both Missouri and Dc, you can file the federal return joint and can use that return to create the state returns (which save you the step of creating a "mock" separate federal return and also allows you to e-file the state returns).