What are the languages associated with each Buddhist school?

If wishing to abound suffering it's only necessary to learn, think and speak and act Dhamma (in simple terms) of which can not be learned in schools or without dependency but only in seeking refuge and associated with those living the heritage. Don't waste your time. Those who wasted their time in first learning, first gain, lost all their accumulated for another time. Once speaking and living Dhamma, if then given, one might use given. Others, by delusion, by following peoples with views, just become this or that kind of "culture-thieves", "dhamma-traders" and on it's cause just long bond slaves, get or stay caught in "marxist-circles".

[Note: this is a gift of Dhamma, not thought for trade, stakes, exchanges or other gains subject toward decay and should be deleted if it's not giften to give in Dhammic conditions]

Language learning is hard, what you should do depends on you. The effort it takes to learn a language is colossal. If a language seizes you, then learn that language. Life is short, you are unlikely to gain competence in more than a handful of ancient languages. On one hand they are easier because in a domain constrained set of texts like Buddhism total vocabulary is small, on the other hand they are harder because humans learn best when they can immerse themselves in a living speaking community, which exists for Pali and Tibetan. Chinese is an interesting case in that literacy in modern Chinese will let you nearly read ancient Chinese without the same effort it would take an English speaker to read Old English.

Professional work. Professionals, like Jan Nattier and the like, are learning ancient Chinese, Sanskrit, Pali and Tibetan, because the sutras often are preserved differently in each and must be compared to recover the original.

If you are not a professional philologists, the English translations are increasingly good enough, there are plenty of concepts I'd have no chance of, nor time for understanding if it were not for modern English commentaries.

There is the special topic of chanting sutras and mantras. In some specific schools, the power of a text depends on it being recited in the archaic language and pronunciation. For English speakers, this creates interesting challenges, because, say in the case of Shingon, the mantras are Japanized pronunciations of Chinese pronunciations of Sanskrit. In Lotus Sutra traditions, there is similar situation where US adherents chant an English accented, Japanized version of the Chinese.

What you describe is very much the path I traveled myself. I studied Literary Chinese and Japanese to be able to read the original Ch'an and Zen texts and I haven't ever regretted the journey. There are so many concepts that are very hard to translate, and often the words chosen for translation are loaded with cultural and religious connotations that blur the message or need to choose one translation from several possible (underlying) meanings. And besides being able to understand the original texts, you also gain a deeper appreciation for the culture from which it stems or where it blossomed.

While it is true that the largest Vajrayana lineages are Tibetan, it is not your only option. Vajrayana also spread to China during the Tang Dynasty and to Japan in the Shingon school. The Taisho Tripitaka is the largest surviving volume of texts for these lineages.

For Mahayana, Literary Chinese would be my first recommendation for the sheer volume of traditions, texts, and commentaries.

Is it absolutely necessary? No. Is it rewarding? Oh my, yes.

Good fortune on your journey!

Question: For each school of buddhism, what would be the best languages to learn in order to be able to study the bulk of the canon associated to that school in its original and why?

Learning Buddhism is like learning Judo. While it might help to know the meaning of some Japanese terms like Ne-waza, Tachi-waza, Nage-waza, etc. In no way it's a mandatory prerequisite for one to become a Judo master. One'll do just fine to know how to practice Ground techniques, Standing techniques, and Throws techniques. If language was the prerequisite then only Japanese fighters could win Judo tournaments (actually, a lot of them do, but not because they're Japanese, but because they really train their butts off). Similarly for Buddhism, it might help with some Sanskrit or Pali, but ultimately what really separates the noble disciple from the run-of-the-mill simply comes down to the level of dedication of one's time and effort into the cultivation of the Path. No shortcut, no secret obscure language, just pure sweat and tears (maybe no tears... :-) )