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I see this has gotten no responses yet, so I’ll try to give a little insight even though I don’t think I can answer all your questions.
There’s gematria and there’s gematria. Anyone can make up a gematria- I’ve been to plenty of Bar Mitzvas and Sheva Brochos where the gematria of the boy or couple’s names are used to link to something like
“mazel tov” etc. It’s nice thought, makes a nice speech. It can be used as a tool to inspire someone or make a point. But I doubt it is holy. But there are gematrias that are brought down by chazal or great Rabbanim that obviously have more weight and significance.
No one knows why Hashem is doing things, even if they have a great gematria to back up their opinions. One is supposed to always learn from the happenings around them, try to figure out what personal lesson can be derived from world events and be inspired to do Teshuva, but no one cay say for sure, Tragedy X happened because person/people Y did Z. One can and should use world events as an impetus to inspire people to improve, but we do not have prophecy today, only great yehidim (individuals) come close to having Ruach Hakodesh (Divine Inspiration), and I would be wary of anyone claiming that they know exactly what Hashem is thinking.
Gematria can be used as an emphasis or hint for many different idea, I don’t think it is a specific tool for the type of diagnosis you are referring to.
Some examples brought down by chazal/our rabbanim:
Tzitzis ציצית in gematria is 600, add the 5 knots and 8 strings and you get 613= wearing tzitzis is to remind you of all 613 mitzvos. (Rashi, quoting Medrash Bamidbar Rabba, Bamdibar 15:38)
Yaakov told his sons R’du, go down to Mitzrayim, and not just Lechu, go. Rashi, quoting the Medrash Rabba, states that R’du רדו in gematria is 210, as a hint to the 210 years of slavery in mitzrayim. In this case it was prophetic.
We don’t eat nuts on Rosh Hashana because Egoz אגוז (17) is gematria of Chet חטא, sin (actually 18, in this case you have to leave off the aleph, which is not heard in the word, so it comes out to 17)- this is symbolic that we should avoid sin.
we find by the “presents” that the Nissim brought to the MIshkan at its inauguration many gematrias that gave symbolic meaning to the present and/or its quantity- ie a silver bowl, קערת כסף 930 – symbolic of Adam living 930 years. etc- see Rashi on Bamidbar 7:18-19, quoting the Medrash.
None of the gematrias were used in the way you are describing.