Interesting question. To answer your question: yes, when reporting imperative speech you should use the present tense in the same way it would have been spoken:
Mother said don't trust strangers.
But I don't think this has do with the imperative form, but rather it's more of an informal way to include reported speech. For example, this paraphrase of a famous line from "Macbeth":
Shakespeare said life is tale told by an idiot, full of sound and fury, and signifying nothing.
I could say "was" but I prefer to use "is" because that's the tense Shakespeare uses in the play. In the same way your example is "don't" trust strangers, and not "didn't" trust strangers, because that's not how someone phrases the imperative. I write it as if I was directly quoting them.
The challenge here is that it's not necessary to always back-shift reported speech into into the past tense, even if you heard it in the past. For example:
He said he is the best, but I don't believe it.
John's sister said they have guests at their house.
Everyone said this election is the craziest they've ever seen.
What these all have in common is that the reported speech could apply to something that is still true. Life could still be an idiot's tale, even if Shakespeare died many centuries before I ever heard Macbeth's opinion.
Which, unfortunately, might make you think that if I do use the past tense, whatever I'm talking about is no longer true. Sadly, that's not the case. English is, frequently, woefully, imprecise:
John's sister says they have guests at their house (The guests are still there)
John's sister says they had guests at their house (The guests have probably left).
John's sister said they have guests at their house (The guests are probably still there).
John's sister said they had guests at their house (The guests might have left, but might still be there)
Usually you can get more information from context, but sometimes you just have to ask someone to clarify what they mean.
Anyway, again, when reporting imperatives do use the present form -- but there is no should about using the simple past with reported speech. It's an option, but it's not required.