There are several issues here. First, let's review the “meaning” of Gegege; then let's think a bit more carefully about what does it mean to try to classify words as "parts of speech", and how does Gegege fit into them.
The word Gegege was created by Shigeru Mizuki based on the way he mispronounced his name as a toddler: Gegeru. So our first attempt could be to classify it as a noun, and intuitively it seems to fit. However, at the same time, it does have the general look-and-feel of a Japanese onomatopoeia, and doesn't seem to refer to something, like "Shigeru" refers to the author; and, in the iconic first-generation anime opening, ge-ge-ge appears to be sung by frogs. Given that the Japanese frog onomatopoeia is gero-gero, it's a fair bet that we can consider it to be an onomatopoia—or, more generally, a sound-symbolic word (also called “ideophone”, “phonomime”, “mimetic word” etc. etc.; Japanese: 擬態語 gitaigo)—that is, a word where the sound of the word itself suggests a feeling, manner or sensation (auditory or otherwise). As a sound-symbolic word, to me at least Gegege feels croaky, critter-y, monster-y, perhaps somewhat yucky (cf. ゲー , a vomit onomatopoeia).
Now, what is it that you want to know when you ask what “part of speech” or “grammatical category” it is? Traditional parts of speech terms, like “noun” or “verb”, have two different and independent usages. The first is formal: it tells how the word relates to other words, more or less mechanically; how it inflects (or doesn't), where can it be inserted… For example, in English, only a noun may follow “the”:¹ you can say “the frog” but not “*the sing” or “*the of”. In Japanese, only a nominal (体言 taigen) may be followed by no. So Gegege is a noun, because it's followed by no; end of story.² If the title was Gegege na Kitaro, then Gegege would be a na-adjective; if it was Gegegeru Kitaro it would be a verb, etc. So we define noun “distributionally”; that is, based on how “noun” words are distributed in actual use among other words.
Please notice that this definition of "parts of speech" has nothing to do with meaning. It treats words as abstract pieces in a game board, and identify word classes by what rules they follow when combining with each other.
A different, older, and fuzzier definition of parts of speech is based on meanings: for example, nouns are "things", adjectives are "attributes" of the things, and verbs are "actions" ("to kick"), "events" ("to die") or "states" ("to like") of the things. This makes intuitive sense, but it's easy to see that this definition is squishy and not very clear:
- A bomb exploded yesterday.
- There was an explosion yesterday.
The verb "exploded" and the noun "explosion" are referring to the same event. (Many nouns refer to events, including "kiss", "death", seppuku and so on.)
- I only kiss hunks.
- I only kiss them if they're manly.
Here the adjective "manly" points to the same thing as the noun "hunk".
We can avoid this problem if we say that nouns usually or typically refer to "things", verbs to "events", and adjectives to "attributes". What's more, if we want to use a noun as a property or an event, we may have to add extra stuff to it: "man" → "manly", "manning" etc. But we never need to annotate or "mark" a noun to denote a thing, a verb an event, or an adjective a property. This insight is due to Croft, who distinguishes (formal, distributional) nouns, verbs and adjectives from (semantic) Objects, Events and Properties; the former typically denote the latter, but not necessarily.
It should now be clear that Gegege is, formally, a noun; but that doesn't mean that, semantically, it must refer to an Object (and it doesn't; it's sound-symbolic). -ne is an entirely different beast; you can't say *ne no Kitaro, *ne da etc.; it always comes at the end of phrases, so it's a phrasal postposition (literally something "positioned after phrases"; Japanese: 終助詞 shūjoshi). And, semantically, it's a discourse helper, comparable to English tag questions like "isn't it?"; it doesn't refer to things, but rather express the speaker's attitude to what they're saying, or the discourse modality.
1) You can also say things like "the beautiful people", where an adjective follows "the"; but this is because the sequence [beautiful people] works, for all intents and purposes, just like a noun, and can be used wherever a noun can ("they are the Englishmen" → "they are the [beautiful people]"), etc. It's a "noun phrase".
And what about things like "the good, the bad and the ugly", which have no apparent nouns? I think you'd agree that, intuitively, it feels like putting the word "ugly" after a "the" makes it into a noun, precisely because "the" expects a noun.
2) There's a small set of strange words which may take no but are not nouns, like kanari "considerably" or tobitobi "here and there". We can identify those distributionally because, unlike nouns, they don't occur as a subject marked by -ga or as an object marked by -wo. I set aside the trickier cases for simplicity. Incidentally, it's easy to find ゲゲゲ as a standalone noun (marked with を／が) on the Internet; but in those cases it seems to be a simple abbreviation for 『ゲゲゲの鬼太郎』.