Are we allowed to use Microsoft SQL Server 2016 Express Edition free of charge in a commercial environment?

Yes, productive use is allowed. However, the Express Edition is an edition that is limited in its scope of use.

It is to be used productively only with the following restrictions:

• 1 processor or CPU
• 4 cores
• 1 GB maximum memory utilized (per instance of SQL Server Database Engine)
• 10 GB maximum relational database size

Source: Official Microsoft Website, as of March 2017 (https://www.microsoft.com/en-us/sql-server/sql-server-editions)