There are multiple parts to your question, but we can focus most of it by restating the main question:
What is ごめんなさい? What does this derive from?
Let's dive in.
ごめん: derivation and part of speech
ごめん itself is 御【ご】 (honorific prefix) + 免【めん】 ("forgiving, allowing, excusing").
The 御【ご】 prefix is a hint that the following piece here is treated as a noun: verbs and adjectives do not express honorific senses through prefixing like this.
Many nouns can be turned into verbs (or verb phrases) by the addition of the basic verb する (literally "to do"), such as noun 勉強【べんきょう】 ("studying") + する ("to do") → 勉強【べんきょう】する (literally "to do studying", idiomatically "to study"). We can do that with 御免【ごめん】 as well, as 御免【ごめん】する ("to forgive, to allow, to excuse").
The なさい part is very common in various expressions, and its derivation is a bit complicated.
なさい is a contraction of older polite なさいませ, the imperative of なさいます, from older なさります, from なさる, from なす.
- なす → "to do", very old word, basically the transitive / causative counterpart to なる "to become".
- なさる → verb なす + passive / honorific ending -(r)aru. This is the older version of what we see today as -(r)areru. The -(r)aru ending used the so-called 四段【よだん】 or "quadrigrade" conjugation paradigm in Old and Classical Japanese, so called for the four different vowels that appear on the verb stem before the various endings (like -masu or -nai), the precursor to today's 五段【ごだん】 or "quintigrade" conjugation paradigm. When adding the polite -masu ending, the verb stem ends in -i, so we get:
- なさります → honorific なさる (showing honor to the actor, the agent of the verb) + polite ending -masu (showing deference to the listener or audience). In certain specific phonological contexts, -ri- becomes just -i-, so we get:
- なさいます → honorific polite なさります + sound change. Then:
- なさいませ → honorific polite なさいます + imperative. Then this contracts:
- なさい → なさいませ dropping the polite imperative -mase. We see this same contraction in the everyday word ください, which also was previously くださいませ (and sometimes you'll still hear this in hyper-polite contexts).
This has two basic uses: one as a replacement for する, and one as a verb supplement / supplementary verb / auxiliary verb (補助動詞【ほじょどうし】, for which the English terminology is a bit unsettled).
As a する replacement
As we saw above, the なす core verb underlying なさい basically means "to do", and as such, なさい can be used as a replacement for する -- or more specifically, since なさい is specifically the imperative conjugation, as a replacement for しろ. So in any case where you have the
[NOUN] + する construction to verb-ify the noun, you can replace the imperative しろ with なさい.
- Note that なさい is from the honorific polite imperative form なさいませ, and from this background, なさい today is still much more polite than しろ.
- Note too that, in idiomatic Japanese, なさい isn't actually used for just any
[NOUN] + しろ construction. For instance, 勉強なさい does not sound natural. For most such verb-ified nouns, natural Japanese generally uses
[NOUN] + しなさい instead -- which leads us to our second use pattern for なさい.
As a verb supplement
なさい is also used as basically a verb suffix, with similar mechanics as we see with ～ます or ～ながら: you conjugate the verb stem into the so-called 連用形【れんようけい】 (for vowel-stem verbs, this is the form ending in -i, same as for the -masu form), and then you stick the なさい on the end. The meaning is the same as above -- this forms a polite imperative, much politer than the plain imperative conjugation, such as しろ above, or のめ for "drink!" or いけ for "go!".
So for 見【み】る in your question post, the なさい in 見【み】なさい isn't replacing any する -- instead, the なさい is used here as a supplementary verb suffix.
Side question: "Can any noun become a verb by adding する?"
Technically, syntactically, yes, that seems to be possible. However, semantically, in terms of meaning, many such combinations just don't say anything sensible. Consider 犬【いぬ】する ("to dog"?) or リンゴする ("to apple"?) or 机【つくえ】する ("to desk"?).
These are syntactically possible, but they produce nonsense. I suppose it's possible that these might make sense, in some super-specific slang-y context, but without that context, these are gibberish.
Please comment if the above does not answer your questions.