"agua" doesn't break the rule. It's feminine. agua fría.

However, there's another rule that words beginning with an a use el because of the double vowel sound.

That's almost correct. It's not all words starting with 'a', but words staring with a _stressed_ 'a', like "águila", "aula" and "hacha" that use "el" as an article, despite being feminine words. The reason is analogous to the use of "an" in English instead of the article "a".

"Abuela"? "Azúcar" "Abuela"?

Lots of little mnemonics, like man creates el problema, the woman la solution.

Those are both pretty easy, too, because they follow regular rules. "-ma" is always masculine. "-ión" is always feminine.

Actually, “rama” (branch), “cama” (bed), “mucama” (maid) are all femenine.

“Centurión” (roman military rank) and “esturión” (fish) are masculine

Correct. "tema", "teorema" are masculine because they are cultisms, derived directly from greek. Same with "fantasma" and "miasma".

bad 'reason': problema is greek too ⲡⲣⲟⲃⲗⲏⲙⲁ 🇬🇷 problem

And it's masculine too in Spanish. Isn't it in Greek?