Firstly, the defending tower and the attacking catapult take their turn simultaneously with the fastest unit in the army, so it is possible for either one to not take its turn, and it is impossible to damage the catapult in any way; player control over them is also impossible. The Ballistics skill only improves the catapult's performance; the attack and defense parameters of heroes affect the damage dealt by towers. There is a bug (or feature) with the Shield spell - towers deal increased damage to units under this spell.
Secondly, towers always take their turn first, then the catapult takes its turn. The catapult has 1000 health points and can be destroyed. The Artillery skill of the defender allows control over the towers, and the Ballistics skill of the attacker allows control over the catapult. There are living siege engines - cyclops; the attack and defense parameters of heroes do not affect the damage dealt.