601.2. To cast a spell is to take it from where it is (usually the hand), put it on the stack, and pay its costs, so that it will
eventually resolve and have its effect. Casting a spell includes
proposal of the spell (rules 601.2a-d) and determination and payment
of costs (rules 601.2f-h). To cast a spell, a player follows the steps
listed below, in order. A player must be legally allowed to cast the
spell to begin this process (see rule 601.3). If, at any point during
the casting of a spell, a player is unable to comply with any of the
steps listed below, the casting of the spell is illegal; the game
returns to the moment before the casting of that spell was proposed
(see rule 721, "Handling Illegal Actions").
As per the rules, do you ever cast a spell and wait for it to resolve before you tap your mana? or does the spell only go on the stack when you pay for the mana?
The rules clearly say that you need to meet all of the casting costs, so if you want to cast Final Payment you need to have all of the costs to cast it, this means, if your life is at 4 an you control no creatures you are unable to cast it, even if your opponent was to take over your turn (by the means of cards like emrakul, the promised end or such) and tried to cast Final Payment to for you to lose the game, they would be unable to unless your life was at 5 or you controlled a creature.
There is no time between announcing that you want to cast a spell and paying for the spell costs.*
This is also the reason why if you cast Final Payment and while it is on the stack you cast the expansion side of Expansion // Explosion you would copy Final Payment without having to pay the additional cost, this is because the card is already on the stack and all casting costs have been met, and copying it is the same as copying any other such spells.
This is the same for any other copy effects.
*There is a window of time there, to activate mana abilities, this is what allows you to use Llanowar elves ability without having to do it before announcing that you want to cast the spell and to tap lands. This is usually disregarded as nothing can be done in this time, but there was a deck that abused this fact in modern not to long ago so I thought I'd mention it.