"that that"?!? how to properly express doubt about a distance object?

There is nothing wrong with "that that."

Now, there is something wrong with saying, "I didn't knew...." The verb phrase is "didn't know" because the helping verb "did" is already in the past tense, calling for the ensuing verb to be in the to-free infinitive form (i.e., "know," not "to know").

With the above in mind, you would write:

"I didn't know that that stove was installed."

Since the first "that," the "that" that follows the verb and introduces the relative clause, can be left unsaid in English, you could also simply write:

"I didn't know that stove was installed." (i.e., "I didn't know [that] that stove was installed.")

Anyway, people say "that that" all the time. There's nothing wrong or even weird about it. It's extremely common. The pronunciation is a little different, though. When people say "that that," how they pronounce it is "thet that." Nobody ever says the two the same, meaning the first "that" doesn't rhyme with the second "that," which rymes with "cat" and "hat." Instead, the first "that" rhymes with "bet" and "let."