The slow version is very misleading imho. Usually the slow version makes it very clear where one word ends and another begins, but in this case it sounds just like: Jag skriver århundrade (t)sbok. There's a clear pause between århundrade and tsbok.
No-one's fault of course, but very confusing if you rely on the slow version.
No, possessives don't take the article - just like how you wouldn't say "the century's the book" in English.
In the construction i år är det hundens år, the det isn't actually a definite article, but the word "it" - as in "this year, it is the year of the dog".