What's stored in spatial_ref_sys? Should I copy data in this table when migrating to the other server?
No, it comes with PostGIS as a standardized catalog of the spatial reference systems your database can work with. It will be recreated when you CREATE EXTENSION postgis;. In fact, copying the table may cause the installation of PostGIS to fail. PostGIS is supposed to provide that table and the schema. You can see the table documented here.

That said, if you have a custom SRS installed such as the ones provided by http://spatialreference.org you will have to copy those rows over, from the docs,


  Various US state plane spatial reference systems (meter or feet based) - usually one or 2 exists per US state. Most of the meter ones are in the core set, but many of the feet based ones or ESRI created ones you will need to pull from spatialreference.org.