The first example is of the first conditional (not the zero conditional as you labelled it) - a present tense verb in the subordinate clause (the protasis, the if clause) with a future form in the main clause (the apodosis, the result of the condition):
If I stick my hands in the fire, I will get burned.
The next example is of the second conditional - a past tense verb in the subordinate clause and a conditional form (would plus infinitive) in the main clause:
If I stuck my hands in the fire, I would get burned.
This is also the second conditional:
If I were him, I wouldn't buy that car.
The past-tense forms used in second conditionals don't represent actual past time. They represent unreal or hypothetical conditions. The hypothetical conditions are, however, imagined as taking place now or in the future, not in the past:
If I won the lottery, I would go on a cruise.
If I fell seriously ill, I would need to go to hospital.
If you were actually writing about a past hypothetical, you wouldn't use the second conditional. Instead you would use the third conditional, which consists of the past perfect in the subordinate clause, followed by the conditional perfect (would have plus a past participle) in the main clause:
If I had stuck my hands in the fire, I would have got (or would have gotten) burned. (BrE prefers "got" here, AmE "gotten".)
If I had been him, I wouldn't have bought that car.
If I had won the lottery, I would have gone on a cruise.
If I had fallen seriously ill, I would have needed to go to hospital. (BrE prefers "hospital" here, AmE "the hospital".)
However, the nature of the unreal condition expressed by "if I were him" is that if you had been him at the time, you would still be him now. Hence in this case it is equally acceptable to retain the simple past in the if clause - but the main clause must still be in the conditional perfect, giving you a so-called "mixed" conditional:
If I were him, I wouldn't have bought that car.
Past narrative. You asked about how we would refer to conditionals in a story narrated in the past, with reference to this passage:
He had wanted to enjoy some drinks with Yarei and report on the events at the monastery, but if he didn't sell the furs that were piled high in his wagon bed, he wouldn't be able to pay for goods purchased elsewhere when the bills came due.
You're right. We have to account for four situations:
- (1) talking in the present about a present/future possible/potential condition (the first conditional)
- (2) talking in the present about a present/future unreal/hypothetical condition (the second conditional)
- (3) talking in the present or past about a past unreal condition (the third conditional)
- (4) talking in the past about a real/potential condition, i.e. the past-narrative equivalent of (1).
For the fourth case, we again use the second conditional.
Imagine that the story was being narrated in the present tense:
He wanted to enjoy some drinks with Yarei and report on the events at the monastery, but if he doesn't sell the furs that are piled high in his wagon bed, he won't be able to pay for goods purchased elsewhere when the bills come due.
Backshifted for the past-tense narrative, this becomes:
He had wanted to enjoy some drinks with Yarei and report on the events at the monastery, but if he didn't sell the furs that were piled high in his wagon bed, he wouldn't be able to pay for goods purchased elsewhere when the bills came due.
The condition was a first conditional at the time when the man had been making his decisions and facing his dilemma about fur sales. When the story is told in the past tense, the present shifts to the past, the present perfect to the past perfect, and the future to the conditional (or future-in-the-past, which is the same form: would).
Note also these remarks from englishgrammar.org (where the "was" is used to illustrate the fact that when a first conditional is backshifted to a past-time account of a real condition, "was" is correct rather than "were"):
If I was in a hurry, I usually skipped my breakfast.
If she was angry, she would shut herself up in her room.
Best Answer
I'd use "is". The I II III system of conditional sentences is a simplification for learners, and it is easy to find examples that don't fit these.
Here the sentence is weird (what does being rich have to do with remembering things?) but I'd expect "He wouldn't forget it if he were rich" or "... wouldn't have forgotten...were..." for the hypothetical. Or "He won't forget it if he is rich" for the conditional about a future event.