Will certain Database Compatibility cause performance degradation?
All the compatibility level does is tell SQL Server to use the older version of the T-SQL language that was available in that version.  For example if within a SQL 2008 R2 server with the database running in SQL 2000 compatibility mode if you try and access some of the new syntax it won't work.

For example 

select *
from sys.dm_db_index_physical_stats (db_id(), null, null, null, 'limited')


will throw an error in a SQL 2000 compatibility mode database.

As for query execution, that won't be changed by the compatibility mode.