I am stunned that the list does not include "Run CHECKDB" (or at least find out when it last ran).

The free but unsupported RML Utilities released by the Microsoft SQL Server PSS engineers can be used to analyze a Profiler trace. That can tell you what is running and how often.

This would be a good spot to plug Red-Gate's SQL Doc tool (no, I'm not a RG employee!). This might not be free but it is a good way to acquaint yourself with a new database.