May an attribute be an entity as well?
This sounds really confusing, but I'd say that's not possible. However, you could have an attribute referencing a specific entity.
Should a "Designation" be its own entity?
IMO this really depends on how it's handled or what you consider a "Designation". If there's a strict 1:1 relation between an Employee and a Designation, then it's quite obsolete and it could be part of the actual Employee. If it's possible to have more than one Designation and/or a Designation might be reassigned/moved/swapped, then yes, it should be its own entity IMO.