Hi,
I'm not sure about the expression, "content that is relevant." I try to make all my content relevant, but sometimes fail.
Often, the status of a condition or situation as to permanance or on-going-ness is common knowledge, or obvious, and sometimes it's not known to the listener or reader (as in the case of your two examples). In my opinion, the writer is free to use whatever tense is appropriate to express his intention, as long as it doesn't present an impossible situation.
I suppose in the case of reported speech, one may ask whether or not the reporter is justified in assuming that a condition is ongoing. The original speaker may no longer have a house in Chicago, and/or may no longer care to play chess. The scenario you describe in
3. is ample justification for using present tense in the
that clause.
Personally, I think the reporter should be allowed to judge whether or not a condition is ongoing. If he doesn't know, and if he's reporting in past tense, he should use only past tense in the
that clause.
He should also be allowed to report present speech about a past condition: "He says he was living in Chicago at the time of the murder." "He tells me his best event was always the javalin throw."
He should also be allowed to use past tense in reporting a permanent condition.
He says/said that to the best of his knowledge, the twin towers are/were still standing. (All four are possible in the right situations.)
He says/said he always thought that Texas is/was larger than Alaska. He says/said Texas is larger than Alaska. He said Texas was larger than Alaska. NOT He says Texas was larger than Alaska.
2. utterance:
I want - He said he wants to go.
utterance:
I will - He said he
will/would go.
Best wishes, - A.