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.
完了 can have 3 different meanings/uses.
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).
All answers miss an important point: the two words are total two different stuff at all! 完 is a verb while 了 is an auxiliary word.
完 in case of this question is used as a secondary part of a compound verb, it denotes the result of the dominated verb. Such kind of compound verbs are called Resultative Compound Verbs. In most situations, say 写完了报告, 了 is used with the whole compound verb instead of just 完. They are not connected as they seem!
The other answers give some good examples, so I'll try to provide an answer from a grammatical perspective.
完 can be used as a resultative complement meaning "to finish doing something". You put it after a verb to mean that you finished doing that verb. 「吃完」 means "finish eating".
了 has many different uses, but the one that I first learned was as a past tense marker. 「吃了」 means "ate".
Now, to talk about something that you finished doing in the past, you combine the two. 「我吃完了」 means, "I finished eating."
There's a ton more detail to learn with these two words. I suggest you look at the Chinese grammar wiki complements page, the completion with 了 page, and maybe the many uses of 了 page. I expect most beginner or early intermediate textbooks will cover these topics as well.
I would translate 完 as "complete," and 了 as "done." 完了 would then be a very emphatic "completely done."
没 and 完 can be used together to indicate an action that has yet to be completed in this pattern 没 + V + 完 for example:
To mean I haven't finished watching/reading it or I haven't finished eating (it)
You can also combine the two in this case it would be:
To mean the same as above.
However the following:
This means I haven't seen/read (it) or I haven't eaten (it)
I (still) haven't read this book or He/she hasn't eaten their ice-cream.