The copies that Kalamax's first ability creates are created on the stack, so they're not "cast." Abilities that trigger when a player casts a spell won't trigger.

You can't choose to pay any additional costs for the copy created by Kalamax's first ability. However, effects based on any additional costs that were paid for the original spell are copied as though those same costs were paid for the copy too.

If the spell that's copied has damage divided as it was cast, the division can't be changed, although the targets receiving that damage still can. The same is true of spells that distribute counters.

Kalamax's first ability considers the entire turn. If you cast an instant spell before Kalamax becomes tapped, its first ability won't trigger when you cast your second.

If an effect copies an instant spell multiple times, Kalamax's last ability triggers that many times.

A copy is created even if the spell that caused Kalamax's ability to trigger has been countered by the time that ability resolves. The copy resolves before the original spell.

An ability that triggers when a player copies a spell resolves before the copy. It resolves even if that spell is countered.

If Kalamax becomes tapped as a cost to cast your first instant spell in a turn, its first ability triggers.

If an effect copies a card rather than a spell (such as that of God-Eternal Kefnet), this doesn't cause Kalamax's last ability to trigger.

If the spell that's copied is modal (that is, it says "Choose one —" or the like), the copy created by Kalamax's first ability will have the same mode or modes. You can't choose different ones.