'cuối cùng' is more like 'the last thing/last time of a series of something' , whereas 'cuối ' can also mean the end part of something. Adding 'cùng' kinda clarifies it a bit. You can also say 'con chim cuối của tôi' but it's more informal
The order of the modifiers after a noun is: attributives -> demonstratives -> possessives.
Unfortunately Duolingo at the moment doesn't actively teach these grammar rules, so you have to look somewhere else to understand them.