SQL server compress backup for lower version databases
I searched on web, but could not find any such stuff which clarifies on what point should we or should we not change the compatibility level of the migrated db's.


As a good practice and suggested by Microsoft you should change compatibility level of database to match with server version on which it is hosted. There are few legacy applications which might require version as 80 for that you have to test, I cannot tell upfront whether you change it or not. You should see what all features were deprecated in 2008 R2 as compared to SQL Server 2000 below two articles would help you

Breaking Changes to DB feature in SQL server 2008 R2

Deprecated database engine features in SQL Server 2008 R2

Deprecated SQL Server features in SQL Server 2008 R2

Regarding changing compatibility level if you face any issue you can immediately change it back to 80, that is why i said a round of testing is required. If you read Change compatibility Level of database it would really help you


  For the Migrated DB's having compatibly 80, are we good to take compressed backups?


Yes you can take compressed backup of 80 compatibility database hosted on SQL Server 2008 R2 subject to condition SQL Server edition is Standard or higher edition