That a very strange story, and dare I say it, it doesn't make a lot of sense. I think I've seen most common Latin textbooks that are used in introductory courses (there aren't that many) and none of them uses "invented" words. I also have my doubts about having to translate Greek mythology texts from Latin, that is highly irregular, especially for introductory courses and I am not aware of a curriculum or a collection of texts that would commonly be used for that.

Secondly, you are wrong in that Latin is a dead language.
A surprising number of people speak it actively and Latin is still the main official language of the Holy See. They even publish a Latin dictionary with modern vocabulary that contains words like smog, gangster, radar, telenovela and hand grenade ("Lexicon recentis Latinitatis"). There are other modern Latin dictionaries available online as well.

There is a chance that your "bicycle" word came from that dictionary and your teacher just didn't know that.

These words are no less made up than the word "computer" is made up in the English language (of course, computer also comes from Latin, "computare", to count). I mean, someone called it that or in some countries groups of linguists come to the opinion that this is what a thing should be called, and voilà, you have a new word. Why shouldn't that happen in Latin?

Latin also was an official language in Hungary, Croatia and Poland until the mid 19th century, so it wouldn't be a surprise to find some fairly modern vocabulary in it as well.

As for the usefulness of Latin today, even if you're not in the Vatican or speak it for fun: It does give you a surprising leg up in learning Italian, French, Spanish, Portuguese, English and basically all technical terms found in the fields of medicine, botany and many more. About 60 percent of English words are borrowed from Latin (or Greek), in scientific or technological areas this number rises to more than 90 percent. About 10 percent of words are purely Latin, the rest found their way into English via French and other influences. Of course that helps you both ways... you will be able to recognize many Latin words without much effort and this can make learning Latin a lot of fun!

That's for the vocabulary only.

As far as what you can read when you study Latin: It opens a whole new world of thought, history, philosophy etc. for you. You will find that in many ways, there where very insightful (and arguably more mature) ways of thinking about things some 2000 years ago. You will also find, to your amusement, that there are several quite popular "self-help books" on the market today which are just badly ripped off versions of ancient Latin texts...

Latin is quite possibly the only language that played a major global role for thousands of years continuously (until maybe the 18th/19th century). That means something. Of course you can read much (but not all) of it in translation (not many people bother, however).

I encourage you, if you're keen, to pick up a better Latin textbook than the one that caused you so much frustration and anger towards Latin. Check out "Lingua latina per se illustrata" for example, which is a simple, amazing book that you will be able to read without doing any translation or learning grammar (much like immersion). Try to translate the title I just mentioned, I'm sure you can understand every single word of it even without knowing any Latin! Well, now you do know some Latin! ;-) It's fun, I highly recommend it.