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Hi. In a section called "Advanced Points in Reported Speech" in the LoreGate.com Learn English Easily site, these two things (exceptions) are what are written under the heading of "exceptions," in addition to one other exception.

Backshift of Simple Present is optional if the situation is still unchanged or if you agree with the original speaker.

Simple Past and Past Progressive do not normally change in sentences with when / if.

Could they mean this?

He said/reported/suggested/mentioned Jane is living with her parents and going to school full- time. -- Would this be correct if someone reporting it knows for sure (how about to the best of his/her knowledge?) that Jane is living with her parents and going to school full-time at the time of reporting?

How about this?

He said they give tests three times a week. -- Would this be correct if someone reporting it knows to the best of his/her knowledge that they give tests three times a week?

He said/reported/suggested/mentioned if Jane came home, he would go playing soccer on the field. -- Does the latter-mentioned exception say that only conditional sentences in simple present and past progressive need not be changed? Does the latter exception mentioned previously mean the sentences don't need to have simple present or past progressive tenses in both if and main clauses of the sentences?
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This kind of technical analysis is not my greatest forte, but I'll have a go at answering anyway.
Anonymous
Could they mean this?

He said/reported/suggested/mentioned Jane is living with her parents and going to school full- time. -- Would this be correct if someone reporting it knows for sure (how about to the best of his/her knowledge?) that Jane is living with her parents and going to school full-time at the time of reporting?

How about this?

He said they give tests three times a week. -- Would this be correct if someone reporting it knows to the best of his/her knowledge that they give tests three times a week?

Yes, that all seems right to me.
AnonymousHe said/reported/suggested/mentioned if Jane came home, he would go playing soccer on the field.
I'm not sure this is the best example because the scenario here seems likely to be:

Actual words: "If Jane comes home..."

Reported: "He said if Jane came home..."

Perhaps a better example would be:

Actual words: "If I left home I'd have nothing to live on."

Reported: "He said if he left home he'd have nothing to live on."
Anonymous-- Does the latter-mentioned exception say that only conditional sentences in simple present and past progressive need not be changed?

Your earlier quote was "Simple Past and Past Progressive do not normally change in sentences with when / if." That part doesn't mention the simple present. I assume the original quote is correct and the above is just a slip.

This is a possible counterexample (tense doesn't change, but situation doesn't seem to be covered by the stated exceptions):

Actual words: "If things had been different I would have stayed."

Reported: "He said if things had been different he would have stayed."

Comments  
Hi. You said in your response:

Your earlier quote was "Simple Past and Past Progressive do not normally change in sentences with when / if." That part doesn't mention the simple present. I assume the original quote is correct and the above is just a slip.

Yes, it was a slip. The original quote is the correct one.