I may be jumping in a bit early, as your post in its current state is a bit confused. However, I think I understand enough of what you're trying to ask to attempt an answer.
I'll restate your core question as I understand it.
How does する conjugate into the negative form しない?
As you note, する is an irregular verb. So things get a bit messy.
Conjugation types: regular "consonant-stem" verbs
In your approach, it looks like you're applying the simplified rules for so-called "consonant-stem" verbs, verbs where the stem (the part that never changes) ends in a consonant. These are verbs like [書く]{kaku} ("to write") or [流す]{nagasu} ("to flush something, to make something flow"). In a lot of English-language materials for learning Japanese, the conjugation patterns for these are described a bit like the following:
Plain form: kak-u nagas-u
"-masu" stem: kak-i- nagash-i-
"-nai" stem: kak-a- nagas-a-
"-ba" stem: kak-e- nagas-e-
Command form: kak-e nagas-e
Volitional form: kak-ō nagas-ō
In Japanese-language materials, verb stems aren't classed as "consonant stems" or "vowel stems", but instead they're classed by the number of different vowels that happen. In the above, there are five vowels, so these verbs are called [五段]{ごだん}[活用]{かつよう} ("five-step conjugation") verbs.
Conjugation types: the irregular sa verb
But how did the (root s•) become (shi し) in the plain negative?
Again, する is an irregular verb: it has no "root s●". 🙂
The Japanese-language name for this specific verb, which has its very own unique conjugation pattern, is サ[行]{ぎょう}[変格]{へんかく}[活用]{かつよう}, or "sa-column irregular conjugation", often abbreviated as サ[変]{へん}. It's the only verb with this conjugation pattern, so you just have to memorize it. It behaves a bit more like a so-called "vowel-stem" verb, only a bit wonky (more on that further below). Here's the pattern for modern verb する.
Plain form: suru
"-masu" stem: shi-
"-nai" stem: shi-
"-ba" stem: sure-
Command form: seyo (Western / Kansai), shiro (Eastern / Kantō)
Volitional form: shiyō
So the negative of する is that stem ''shi-'' plus the negative ending ''-nai'' = しない.
History
In both modern and Classical Japanese, the verb する is classed as サ[行]{ぎょう}[変格]{へんかく}[活用]{かつよう}, or "sa-column irregular conjugation". In Classical Japanese, the verb had the following verb stems (the part before any conjugation suffixes like ～ない or ～ます). For comparison, I've added the s-column endings for the Classical [二段]{にだん}[活用]{かつよう} ("two-step conjugation") verb, one of the patterns that evolved into the modern "vowel-stem" verbs, as well as the s-column endings for a Classical [四段]{よだん}[活用]{かつよう} ("four-step conjugation") verb, the precursor to our modern regular [五段]{ごだん}[活用]{かつよう} ("five-step conjugation") verbs. I'm also including the Japanese names for each conjugated stem form. (Note that the "attributive" replaced the "conclusive" in modern Japanese.)
- [活用]{かつよう}: サ変 二段 四段
- [未然形]{みぜんけい}： せ せ さ mizenkei, "irrealis" ("hasn't happened yet"): -nai stem
- [連用形]{れんようけい}： し せ し ren'yōkei, "infinitive": -masu stem
- [終止形]{しゅうしけい}： す す す shūshikei, "conclusive": plain form
- [連体形]{れんたいけい}： する する す rentaikei, "attributive" (used to modify a noun, like an adjective)
- [已然形]{いぜんけい}： すれ すれ せ izenkei, "realis" ("assuming it has happened"): -ba stem
- [命令形]{めいれいけい}： せよ せよ せ meireikei, "imperative": command form
You can see how する behaves much more like a Classical vowel-stem verb: only the [連用形]{れんようけい} or -masu stem is different.
Classical negative
In Classical Japanese, the negative form of する was せぬ. This is the -nai stem せ + negative suffix ぬ.
Beginners of Japanese usually don't encounter negative ぬ, since Eastern Japanese (Tokyo Japanese, also called Kantō Japanese) is the "standard", and this ぬ isn't part of everyday Eastern Japanese. It's still used to some extent in Western Japanese (Kansai Japanese, as spoken in places like Kyōto and Ōsaka).
Shift to the modern negative
The shift from Classical negative form せぬ to modern しない included two changes -- 1) the Western and older negative suffix ぬ was replaced by the Eastern negative suffix ない, and 2) the せ shifted to し.
#1 is pretty straightforward, but #2 may require looking at the phonetics (sounds).
In modern Japanese, we're taught that the i vowel is special for the s- kana -- the consonant part of し is pronounced not like s-, but like sh-, as shi. This kind of fiction-y consonant sound is called an "affricate" in phonetics studies. In older times, up through at least the 1600s and possibly later, せ was also pronounced as an affricate, as she, instead of the modern se.
Beginning materials for Japanese often omit any discussion of pitch accent. For する, the negative form in Eastern Japanese has a so-called "flat" or low-high pitch, shown in some dictionaries as pitch type "0" (zero, meaning no downstep) or by marking the word like [しない]{LHH}. This means that the first mora (the し) has a low pitch, then the な and the い have a high pitch. As compared to a word in the English language, it's a bit like the し is unstressed. When し is unstressed in a Japanese word, it's often pronounced without the vowel value, more like just sh without the i. You might have already noticed that して is pronounced more like shte.
I haven't read any academic studies about this change, but I suspect these two factors may have combined: after the shift to Eastern Japanese, せぬ or せない would have been pronounced more like shnu or shnai, making the initial sound ambiguous as to whether it should be spelled せ or し.
Is じゃない even a verb?
This is more of a phrase than a verb.
じゃ is a contraction of では.
- で is sometimes explained as the conjunctive form of plain-form だ and polite-form です, the copular ("to be") verb.
- The は is the same as the topic particle, used here in what is sometimes called the "contrastive は" construction -- "contrastive" because you're actually negating the statement.
ない is the basic negative in the plain form, same as the suffix ～ない in しない, only used independently to negate nouns and ～な adjectives. This ない conjugates the same as any ～い adjective, with adverb form なく, past tense なかった, etc. The polite form for this independent use is ありません, and you might even see ～ではありません in some of your materials as the polite way to say "it is not ～".
