1) Different months are the "first" for different things, just as the secular calendar begins from January but the school year starts in September and a corporation's fiscal year could begin in a different month.
2) The Ibn Ezra explains in a similar vein that Sukkos is the conclusion of the agricultural year, when last year's produce is gathered into the storehouse in advance of planting the next year's crop in Cheshvan.
3) First of all the Gemara refers to Vayikra 25:9 where the word "terua" is equated with sounding the shofar. Secondly, we commonly find the Torah Sh'Baal Peh elaborates on unclear parts of the Chumash. Other examples would be tefillin where the Torah Say to bind "them" on the hand and arm, but doesn't explain what "them" is. Also, the Torah says that one should slaughter an animal "as I commanded you" but we do not find in Chumash any details of how Hashem commanded us to slaughter.
4) There are three opinions if "terua" refers to what we call "terua" "shevarim" or "shevarim-terua", in any event all are very similar crying-like sounds. If anything, the fact that our safek is limited to such a minor detail itself proves that on the bigger issues where there is no disagreement the tradition is certainly intact.
5) The Ramban in Shmos 12:2 writes that the names of the Jewish months were borrowed from the names of Babylonian moths to remind us each time we mention the name of a month that we were in exile in Bavel, and from there Hashem redeemed us and brought us up to our the land of Israel and ultimately Hashem will also redeem us from our current Galus with the coming of Moshiach.
6) The Gemara in Sanhedrin says that the current alphabet was the original one, just it initially was only used for sacred texts while kesav Ivri was used for the vernacular. After Ezra HaSofer returned from Bavel to Eretz Yisroel he instituted that Ashuris be used exclusively.