CC, Deen does not mean 'way of life', rather it means -
religion
creed
faith
belief
Dear Jadz, word DEEN is from root DAAL, YAA and NOON. It has many meanings but one of its meanings in the quranic context is a way of life for mankind advised by Allah as opposed to any way of life people adopt for themselves. This root has meanings you have stated but in addition it also has meanings like debt, loan, borrowing, judgement, law, custom, a code to abide by, constitution, tradition, practice, ruler, judge, controller, debtor, creditor, lender, borrower, punishment, reward, habit, to worship, conviction, slave, to be enslaved, to be owned, to compel, to submit, to own up, to sin, the teachings, obedience, reckoning, counting, calculation etc etc.
Following references may be found helpful.
http://islamawakened.com/quran/1/4/
http://www.islamawakened.com/quran/2/256/default.htm
http://www.islamawakened.com/quran/3/19/default.htm
http://www.islamawakened.com/quran/8/39/default.htm
http://www.islamawakened.com/quran/9/29/default.htm
http://www.islamawakened.com/quran/9/36/default.htm
http://www.islamawakened.com/quran/37/53/default.htm
http://www.islamawakened.com/quran/56/86/default.htm
http://www.islamawakened.com/quran/109/6/default.htm
In ancient languages roots were constructed using sets of letters which in turn were meaningful words presented as picture of things so each letter represents a concrete reality and properties of that reality as well as its uses and whatever purpose it served. Therefore a root ended up having many many meanings as explained by ancient authors of the quranic tafaseer and dictionaries. If one could get one's hands on such dictionaries all this can become very clear. For example, letter DAAL means door, letter YAA means hand and letter NOON means fish or in other words ancient people drew pictures of door, hand and fish in order to represent something we call DEEN. So word DEEN means door, hand and fish as well as anything like them in some sense for some purpose. This is how all languages came about. In short our alphabet sets have letters which were meaningful words that were used to label concrete realities.
Madhab means - amongst many things -
opinion
view
belief
ideology
teaching
doctrine
movement
trend
school
and so on.
Word MAZHAB is from root ZAAL, HAA and BAA. This root also has many meanings like you have stated but also gold, to go, to go away, to set out, totake away, faction, sect, to adopt a conviction, to disappear, to vanish, to go off, to clear off, to go from, to depart from, to leave, to go away with, to claim, to appreciate for oneself, to remove, to cause to vanish, to abolish etc etc. Letter ZAAL is alternate to letter DAAL, Letter HAA means window and letter BAA means tent or house so word MAZHAB has all the meaning present in these letters because each letter presents a concrete reality and its description both in sens eof its properties as well as its actions or uses or sense of purpose involved in its use.
Islam means
submission (to the Will of God)
Muslim means
one who submits (to the Will of God)
Word ISLAAM comes from root SEEN, LAAM and MEEM. This root also has many meanings eg peace, ladder, staircase, submission, to be at rest, safety, tranquillity, completeness, being free from obstacles, to become resigned to, to hand over, to recive, to accept, to stroke, finger bones, to give in, to greet, to salute, to spare, to bail out, to rescue, to save, to surrender, to devote oneself, to porfess the deen of islam, to end hostility, to be sound, not threatened, to be pure, to be clear, to be free of impurities or defects or faults, to be whole, to be sick, be heavy, be troubled etc etc. Letter SEEN is altered form of SHEEN meaning tooth, Letter LAAM means ox-goad and letter MEEM means water. So whatever these concrete realities represent are all present in word ISLAAM as its meanings. All we need to do is see what a tooth is, what are its qualities or properties or characteristics or what it does or is used for and how it functions for a any particular use or context.
My explanations will help people see how people brought about languages by use of words and how they gave them concepts and meanings and brought about grammatical rules. It must always be remembered that languages came long before people thought of writing books to explain grammatical rules or gave words meanings and wrote any dictionaries.
Actually if we observe a baby constantly after birth and see how he learns to talk, we can see how original people brought about human language. It was not the case that one day someone sat down to invent a language and after spending some days he came up with a perfect language. It took humanity many generations to take these very simple steps generation after generation.
regards and all the best.