Learn Finnish: Hän pilaa terveytensä savulla. - He ruins his health with smoke.

When a third-person subject acts on their own possession (like 'his own health'), Finnish drops the genitive pronoun hänen and only uses the possessive suffix (-nsä). If you included it and said Hän pilaa hänen terveytensä, it would mean he is ruining another person's health.

Since the health is the object of the verb, shouldn't it be in the accusative/genitive case (terveyden)? Why is there a 't' instead of a 'd' in terveytensä?

This is a tricky rule about possessive suffixes. The normal object form is terveyden (weak grade -d-, ending in -n). However, when you add a possessive suffix to a word that ends in a single -n, the -n is dropped. Dropping the -n opens the final syllable, which forces the word to revert to its strong consonant grade. So terveyden loses its -n, the weak -d- flips back to the strong -t- (terveyte-), and then we add -nsä.

Does the adessive ending -lla in savulla literally mean 'on smoke' here?

No, it means 'with smoke' or 'by means of smoke'. While the adessive case (-lla/-llä) is mostly used for location ('on' or 'at'), it is also the standard case used to express the instrument or tool used to perform an action.