I'll give it a shot.
The first pair is really cool. And tricky. It is the word "just" that seems to throw the whole thing off balance.
I just read that the characters of The Pirates of Mars and Its Infernal Vicinity would be coming to our city.
This is actually correct, if a bit too formal or awkward (choose your poison).
I hasten to add that, upon Alex K's suggestion, I did some major pondering, and suddenly - voila! It dawned on me I was wrong.
It's neither too formal nor awkward, it's quite simply incorrect. The rest (specifically the part about the word "just") applies as before.
I just read that the characters of The Pirates of Mars and Its Infernal Vicinity will be coming to our city.
This is correct because, singularly enough, the word "just" almost succeeds in shifting the first half of the sentence into the present perfect, justifying the "will" in the second half. As if it were "I have just read that the characters of The Pirates of Mars and Its Infernal Vicinity will be coming to our city."
The second pair is less tricky:
The notice board on the gate of the manor read that the house had a dog that we should beware of.
This is correct.
The notice board on the gate of the manor read that the house has a dog that we should beware.
This is wrong. It should be "had," no question.
Addendum: In your story, both the notice and the dog are in the past, no matter how immediate. There is no evidence that either exists in the present: neither the notice, nor the dog. If you wished to shift one into the present tense, you'd have to be polite and extend the same courtesy to the other. As in:
"I was there this afternoon. Upon my word, the notice board on the gate reads that the house has a nasty dog that's easily offended and hates burglars."
