Q2.: Why use two "ifs" in a row?: もし もしも
It's just repetition for emphasis. We do the same thing in English, right?
If...
if that's the case...!
Note that this translation isn't quite accurate because here, this expression leads into the following sentence.
(edit: Whoops, Toritoribeさん already explained this. My bad.)
Well, I know 「ある」can either be an auxiliary verb as part of the ーて construction or just a main verb to indicate that something inanimate exists. It's negative form is 「ない」.
It's the latter here, because it's simply Xがありそう.
Toritoribeさん and mdchachiさん tried to point you in the right direction, but I don't think you've been able to research the difference between あるそう（だね） and ありそう（だね）.
Maybe go back to the Dictionary of Basic Japanese Grammar and look up そう. There should be two entries, one for plain form + そう, and one for stem + そう.
Adjectives can either be prenominal (before the noun they modify) or used in conjunction with 「の」.
Not sure what you mean by "used in conjunction with の". Would you care to elaborate so we can tell if you understand this? Lots of things can be "used in conjunction with の" (and there are many different の's).
「な」adjectives conjugate like nouns (I think it's weird Tae Kim says "nouns" since it's です that's conjugated that way)
The copula だ／です is used to form predicates with nouns.
（私は）裕福です。 (I'm) wealthy.
（私は）裕福じゃないです。 (I'm) not wealthy.
（私は）学生です。 (I'm) a student.
（私は）学生じゃないです。 (I'm) not a student.
Perhaps that illustrates how な-adjectives (裕福） and nouns (学生） conjugate the same way?
そう is used to express a quote: "likely outcome given the situation".
Not sure what you mean by "quote" here. stem+そう (which, again, is different from plain form+そう) can mean "seems/looks" or "seems about to", but this doesn't really have anything to do with "quoting" anything. (If anything, plain form+そう is sort of a quote, or hearsay, but the two are completely different structures. Again, I suggest you research the difference so you don't get tripped up in the future.)
もともと 占いなど
当たらなくて 当たり前なのだ。
もし もしもだ…
"Inherently my predictions and the like
are wrong and [then right, it's obvious/and then right, obviously].
If that's the case..."
If I have this wrong do refer me to certain concepts so I can make my own research, thank you.
Still not sure where you're getting "right" from.
"Wrong, then right" would be something like 当たらなくて、当たって（当たり前） (て form of the negative, て form of the positive, followed by 当たり前) (or 当たらなかったり、当たったりして当たり前, the "inexhaustive listing" たり form which you've seen before in 聞いたりする). This is just 当たらなくて (note: it's probably better to think of this as "the て form of the negative" rather than the "negative of the て form") connected to 当たり前.
Note also that even though 当たり前 uses one of the same kanji, it's not a form of the verb 当たる, but rather a single (and different) word in itself. (You're on the right track, at least, with "obvious".)
（占いが）当たらなくて困った
（占いが）当たってよかった
（占いが）当たらなくて当然だ
Here are some examples of the て-form of 当たる (both positive and negative) connected with different predicates. Care to give a shot at translating them?
3. "for all negatives, the 「い」must be replaced with 「さ」：
Ex.: これはなだの試合じゃなさそうだ。
So figured I'd remove the trailing 「い」and replace it with 「さ」: おいしさ- and then add the そう therefore: おいしさそう. "this is likely delicious".
試合じゃない→試合じゃなさそう is negative
おいしい is not negative.
You need to negate the い-adjective おいしい before conjugating it into the そう form.
(i.e. we're trying to say "this
doesn't look delicious/tasty")
(edited for clarity)