There is no distinction between the preposition, the conjunction and the adverb in English. In Portuguese the adverb is depois, the preposition depois de and the conjunction depois que.
After lunch, we go for a walk.
He told me that, after he had done it.
What comes after(ward)?
* After as an adverb is actually a pronominal adverb. After refers to something. (Except in expressions like to look after, etc.). In this sentence it must be clear after what he will come. Without referencing something it would be later: He comes later.
depois later / after => Adverb
Vou pagar hoje ou o faço depois?
Do I pay today or do I pay later?
depois de after => preposition
Aqui temos algumas dicas de como relaxar depois duma maratona.
Here we have some tips how to relax after a maraton.
depois que after => conjunction
O que perdeu a importância depois que tive um filho.
What lost any relevance after he got a child.
As already has been mentioned very often, see chapter 16, in front of an infinitivo (pessoal) only a prepostion is possible. It has equally already been said the infinitivo pessoal doesn't differ in its form the futuro do conjuntivo. However the futuro do conjuntivo is not a verbal noun and together with a preposition only a verbal noun can be used, because only a verbal noun can have the syntactical function of an object. From all that we can deduce that in the following sentences that the verb form is an inifinitivo pessoal and not a futuro do conjuntivo.
depois de with infinitivo / infinitivo pessoal
Ficou tonto depois de cair do cavalo.
He was dazed after he had fallen from the horse.
Prometo que vou trabalhar nisso depois de morreres.
I promise that I am going to work on it after your death.