How do I replicate SQL Server index INCLUDE and STATISTICS functionality on PostgreSQL?
I don't know what CREATE STATISTICS does, but statistics for the optimizer are collected using the ANALZYE command when autovacuum is running - which is turned on by default.

Statistics are always collected for all columns, no need to turn it on specifically. 

You can control the level of details collected for the statistics on a per-column basis using ALTER TABLE ... ALTER COLUMN column SET STATISTICS integer.

An index in PostgreSQL is always non-clustered, so I'd assume that the above index maps to a regular index on the QuestionId column. 

Not sure about the INCLUDE part. I assume this is to support index only retrievals if that index is chosen by the optimizer.  As PostgreSQL does not yet have an index-only retrieval, there is no equivalent technique there.