名前が草虫であるべきだね
Why doesn’t である take a particle? Is it an exception with this word, or am I missing something? It looks like the で particle is built into the word? But that particle doesn’t make sense to me if that’s the case.
Can you really follow a noun with a verb without the use of any particle?
I think it’s because である isn’t a verb but a form of the copula. You wouldn’t use particles with 「だ」 or 「です」 so you wouldn’t use them with 「である」 either.
But also, in casual speech there’s plenty of times you won’t use particles with verbs.
「先生いる？」
Ahh, I see! Since the word means “is” which is the same meaning as だ right? So it’s basically a modified だ. Thanks for the help!
Technically it’s not a modified だ, though they do serve the same grammatical purpose. である is a more formal or literary version of the copula compared to だ (plain) or です (polite).
Is である used for the same as だ/です? What are the different use cases? である is a copula, but also a verb?
A copula (in linguistics) is a word that links the subject to the predicate. It’s actually often a verb, though not in all languages. It happens that it is indeed a verb in both English and Japanese. (Because of this, it can conjugate like a verb - so you can say だった or でした for example)
You can use である in the same spots you would use だ or です, however it cannot be added on to adjectives or other verbs in order to add politeness like です can. Because of the different registers, you might have to change the sentence so everything is at the appropriate level of formality or politeness.
高村は先生（だ）
高村さんは先生です
高村は先生である
All of these mean “Takamura is a teacher.” だ is often omitted or softened with ending particles when it comes at the end of a sentence like this.
So である is another form of formality like です? So I can say 名前が草虫だべきだね too?
I finally understand that now! Thanks!
Just to add to your explanation, the opposite is not true. And I’m gonna steal OP’s example for that:
名前が草虫であるべきだね
You can’t replace である with だ/です in that case. Basically, whenever you want to modify a noun using another subsentence (this has a proper name, but I don’t know it) you’ll want to use である
Example: 先生である高村… (Takamura, who is a teacher…)
So である can replace だ/です, but である has specific uses too that can’t be replaced by だ/です?
And である can modify the noun before it with what comes after it? 僕であるプロゲーマ. Is this basically just super formal?
名前が草虫だべき doesn’t work then because べき is a noun? Or is it not a noun, and it does work?
This video might help too:
(Obligatory Cure Dolly warning, the android presentation is a little weird, but if you can get past it I find her explanations to be super helpful!)
You got that right 
Almost! It’s backwards: である can modify the noun after it with what comes before it. So your example should be プロゲーマーである僕.
And it’s not just what comes before it that does the modifying, it’s the whole thing, including である.
So プロゲーマーである (is progamer) is modifying 僕 (me) so that it now means ‘me, who is a progamer’ or ‘me, the progamer’ which sounds more natural in english.
Hmm, I don’t know if べき qualifies as a noun or not, sorry. I just know that you need to use である there.
Edit: 僕であるプロゲーマ is technically gramatically correct, but it just sounds weird since it would mean something like ‘the progamer who is me’ but maybe there’s some obscure scenario where it would make sense to say something like that
Funny, I just watched this today.
だ is actually short for である.
One thing that may be tripping you up is that polite and formal aren’t the same thing in Japanese.
高村さんは先生です - polite Japanese 敬語／尊敬語
高村は先生である - formal Japanese 丁寧語
高村さんはであります - polite and formal
It’s called a verb phrase I think. And there’s a general rule of thumb with Japanese that what comes before modifies what comes after.
So a verb phrase in the dictionary form will modify the noun directly following it.
Others have basically answered this but, だ and です cannot be used to modify a noun in a relative clause. That’s why you have to use であるべき and not one of the other copula in that particular grammar pattern.
Edit: to clarify, although I’m not sure if べき is technically a noun since it’s only used as an auxiliary, but it functions as a noun in terms of how other words interact with it.
Thanks guys! This helped me a lot! 
To add to this, the proper go-to keigo for all this would be using ございます instead of あります. You’ll often see ～でございます used in anything customer service.
Aye but I didn’t want to confuse things further. 
TL;DR, mainly for @JesperHH:
Uh… my understanding is that the である体 is actually the written/literary form. It’s not really ‘formal’ unless you mean ‘used in formal writing like an academic paper’. And it definitely doesn’t fall under 丁寧語, because 丁寧語 is just the most common form of 敬語 and literally translates as ‘polite language’. 敬語 is ‘respectful language’, and 尊敬語 is… well, ‘respectful language’ as well, but it’s specifically ‘language that shows respect to a person of elevated status’ because 尊 means something like ‘noble’ or ‘honorable’. That’s why we also call it ‘honorific speech’ in English. However, the use of Japanese technical terms aside, yes, I completely agree with you: ‘formal’ and ‘polite’ aren’t entirely interchangeable, because the first is about respecting conventions/protocol, while the second is about respecting someone else.
OK, so, @alo broke it down for you, and I’d do the same, but slightly differently: when used at the end of a sentence
However, when it comes to using a form of だ・である・です at the end of a relative clause in the middle of a sentence, you only have one option: である. It’s often used for phrases expressing like 王であるアーサー: ‘Arthur, the king’, more literally ‘Arthur, who is the/a king’. It’s a form that links to nouns. (連体形=‘noun-linking form’. I mentioned this on another thread in a reply to you, I think.) です is a polite form of a verb, so it doesn’t have a noun-linking form. だ has a noun-linking form, but it’s very specialised: it’s な, which only appears before のだ and related の structures, and after な-adjectives.
To return to this, yes, である is the copula, but another way you could see it is that it already has a special particle in it: で. If you look at theories on how we ended up with である・だ・です, you’ll see that the idea is that we had なり・にあり at first (あり is from ある, of course), and then someone added て to get にてあり, which became であり・である, and the rest of the forms evolved from there. Even today, にて is still an equivalent of で, but I think it’s quite formal and rare. So in short, another explanation is that the copula carries a particle within itself.
As for why you have to use である here, like I said, you have a relative clause (a phrase containing a verb/adjective and a noun that modifies something else) that describes/modifies べき, so you have to use である even though as a sentence on its own, 名前が草虫 could end with だ, です or である.
It isn’t a noun but…
as @phyro said, it acts like a noun. However, this is only true relative to what comes before it. べき・ベく・べし is actually an adjective – or an auxiliary verb if you use traditional Japanese classifications – that has a modal function and indicates what ‘should’ be done or may be possible. For example, なるべく速くする means ‘to do as fast as possible’, but in a literal sense, it’s ‘to do fast, as much as can possibly become [true]/as [the action] can possibly become’. べく is the 連用形, which is usually the adverbial form for an adjective. Your sentence, meanwhile, means ‘Its name should be 草虫, right?’, but it’s actually a little confusing for me (more on that in a moment). This idea that something can/should happen is why it’s sometimes written like this using kanji: 可き・可く・可し. 可 indicates possibility, and less commonly permission/acceptability/the fact that something is considered good to do.
The reason I say べき an adjective is that its forms change like an い-adjective, which used to end in き:
For comparison, やすい used to have the analogous forms やすき, やすく and やすし.
Another example of an auxiliary-verb-cum-adjective that acts like this is ごとき・ごとく・ごとし. It’s usually modified by a noun that’s attached to it with の or directly. 〜のことき roughly means ‘like ~’.
Now then, why I found your sentence confusing…
名前が草虫であるべきだね makes it sound like the name of something ‘should be 草虫’ because calling it that is the right thing to do according to morals, common sense, social expectations etc. It’s not that it seems likely to be called 草虫, because in that case, はずだ would be used instead of べきだ. Could I ask where you saw this sentence, and in what context? Because it seems strange that calling something 草虫 would be considered a duty/the right thing to do.
So である works in two ways: Ending a sentence and joining two nouns.
And べき works like a noun in this sentence, so that’s why である is used. だ can’t be used to join two nouns, but it’s used with the explanatory の in the middle of sentences? I didn’t quite understand what you meant with the な, can you elaborate? Sorry 
I didn’t really learn any grammar when learning English nor my native language(Norwegian), so there are a lot of new concepts I’ve had to learn recently because of Japanese. The concept of copula is still very new to me. My point is; there are tons of things I don’t understand yet. What does a copula really mean? I don’t think I have a full understanding of it yet.
Relative clause? So 草虫 is the noun, and である is the verb, that modifies べき? Or do you mean that either a verb, adjective, or noun can be used. Or did you mean that you need a adjective/verb + a noun? And does the verb have to be used as an adjective when used in a relative clause?; “たべたうさぎ”.
What verb is it a polite form of? Isn’t である also a verb in that case? Copula in Japanese are all verbs, right? Are だ and である fully-fledged copula, and です just a polite form of だ?
What is なり and にあり? ある is the inanimate form of “to be” right?
I see. So even if I use である in the sentence it doesn’t have to be a polite sentence?
I have never seen its other forms. べく makes it an adverb? べし I’ve got no idea of what this could be though.
Here べき takes on another meaning then?
What do you mean? I’m lacking a lot of vocabulary on these topics
I think I understand somewhat what you’re saying, but there are too many concepts and words I have no knowledge of yet. It’s hard to see how it all ties together, and a lot of information to unpack.
I wrote the sentence myself. A Japanese person didn’t know what grasshopper meant, so I sent her an emoji of a grasshopper. She then told me the Japanese name for grasshopper, which doesn’t contain the word grass in it. That’s why I came up with the name 草虫, and wrote a sentence I believed to mean something like: “It should be named grasshopper”.
Yeah I may have been a bit loose with the definitions and conflated the English terms for polite and formal.
Like @wantitled mentioned, in spoken form it would be でございます.
The point I wanted to get across was that the use of those terms rather than だ/です feels more ceremonial, I guess, although it’s hard to find a word that conveys it exactly.
In practice I think of it like:
丁寧語 - “proper” Japanese (in quotes because I don’t think polite quite covers the nuance I feel)
敬語 - humbling yourself
尊敬語 - elevating the other person
In all cases though, word choice is influenced by these so it’s hard to tease out the actual grammar and if you’re just learning they can muddy the waters a bit.