What is the difference in use between 完 and 了? If I understood correctly, both 完 and 了 can be used to indicate a completed action, so what is the difference in use? How do we properly use 完 in a sentence? Which contexts would be valid?
You can use 完 and 了 together or separately. 


了 is usually used to indicate the completion of an action. E.g. 你买了好多东西 (You purchased a lot of stuff). See the question "Tense and use of 了" to learn more.
完 is used to indicate the action of completing/finishing something. E.g. "說話沒完的人" (a motormouth, someone who talks to no end). Usually it's verb + 完.


完了 can have 3 different meanings/uses. 


Verb + 完了 can indicate that the (verb) action has come to an end. 
Noun + 完了 indicate the feeling of despair or the end of something . E.g., 哎呀，我完了！ (On no, I'm finished/done for!) or 那個車完了(That car is finished/done for!). 
完了 by itself (usually at the beginning of a sentence) is usually an exclamation of despair. E.g., 完了, 车没油了！回不了家了！ (We're/I'm done [for], the car's out of gas. We/I can't go [back] home [now]).


没完 can be used to describe things left undone, or something never-ending/eternal. E.g., 活儿还没完 (Work is not done/finished yet) or "說話沒完的人" (a motormouth, someone who talks to no end).