What types of methods can we call from a .NET assembly in SQL Server
SQLCLR is a very limited CLR host. Yes, you can only call static methods due to the App Domain being shared across all sessions (i.e. @@SPID), and hence any "shared" instances or variables would be very much not "thread safe". App Domains are created on a per-Database, per-"owner" (i.e. whatever User is listed in the AUTHORIZATION clause of CREATE ASSEMBLY, or dbo as the default if that clause is omitted), so there is a minimal amount of separation, but for any given method that is exposed via a CREATE {object} AS EXTERNAL NAME statement, any User in any Session executing that SQLCLR Stored Procedure, Function, Trigger, or User-Defined Type will be hitting the exact same App Domain. This is why you are also not allowed to store values in static class variables (unless they are marked as readonly) as they are shared across Sessions and one Session can overwrite the value that another Session just stored and hence you would get odd / erratic / unpredictable behavior.

Regarding SQLCLR being a restricted environment:


Please see the following MSDN page for a list of what is not allowed in SQLCLR: CLR Integration Programming Model Restrictions.
Please see the following MSDN page for a list of what few .NET Framework libraries are included in SQLCLR: Supported .NET Framework Libraries.
For more information on working with SQLCLR in general, please see the series I am writing on this topic on SQL Server Central (free registration is required to read their content, but it's worth it): Stairway to SQLCLR.