To address some points raised:
(1)
Honorific language is a hodge-podge of verb forms and locutions used to show respect. The fact that the polite past tense of the copula でした is used to create the polite past tense of 読みません is just an accident of historical usage. Originally there were other competing forms: -ませなんだ, -ませんかった, -ませんだった, in addition to -ませんでした. The polite past tense form of the copula is being used only to convey politeness, with no regard to its formal meaning as a copula. One might compare the use of the causative form of the verb to indicate extreme politeness that is common in classical Japanese. To try and force some causative sense from it would be fruitless. And then there is the causative-passive in classical Japanese...
Generally, the longer a locution, the more it is percieved as being polite. This tendency is found in English (at least as spoken by British English speakers) as well, 'Can you close the door please?', 'Would you be so kind as to close the door please?', 'May I trouble you to close the door please?' etc. English can't rely on many special verb forms unlike Japanese.
(2)
だ・です can be split, だ => である です => であります, and then additional material can be inserted. One can say 昨日は学生ではあった・ありました, 'yesterday I was a student'. (There is some kind of implied comparison here, 'yesterday I was a student [but today...]', but the sentence can stand on its own, since context will make it clear.) One can also say 昨日は学生でもあった・ありました 'yesterday I was also a student'. Further examples:
だから中国は専制君主制の国家でながくありましたけれども・・・ 'Thus China was long a county under an absolute ruler, but...
吃音{どもり}で、無口な暴君でわたくしがあれば・・・ 'If I were to be some stammering, speechless tyrant...'
(3)
Since one can find many examples of acceptable Japanese such as
沖縄を本当に愛してくれるのなら
仕事を本当に早める
健康診断の結果を本当に理解していますか。
it would appear difficult to claim that 気を本当に付ける is not acceptable. There is a syntactical operation in Japanese known as 'scrambling' (かき混ぜ操作) whereby sentential constituents such as noun phrases, adverbial phrases, can be moved from one position in the sentence to another. 本当に is an adverbial phrase, and can be freely moved. Reasons for scrambling are probably to do with shifting emphasis around in the sentence. An example:
a. 喫茶店の主人が元町商店街で犯人らしき男を見たそうだ。
b. 元町商店街で,喫茶店の主人が犯人らしき男を見たそうだ。
c. 犯人らしき男を,喫茶店の主人が元町商店街で見たそうだ。
d. 元町商店街で,犯人らしき男を,喫茶店の主人が見たそうだ。
a. is the normal, unmarked order of constituents. b., c., and d., show a constituent moved to the front of the sentence, presumably to highlight that constituent.
The case of 気を本当に付ける is thus nothing more than an example of the above operation.
[Addendum]
I'd like to address another item where there seems to be further, terrible confusion in regard to the use of です.
The following list of sentences, cited above, is suppose to show that treating だ・です as the copula is misleading:
読みたいです。
読むでしょう。
読みませんでした。
読まないとです。
ご主人様はそれをお読みですか。
私はですね、この本を読みたいんです。
本当に?私もですよ。[as answer to the above]
あの子にじゃなくて、あなたにあげる。
As discussed earlier, the polite forms of the copula are used as a suppletive device to form honorific forms of verbs which otherwise don't exist; thus, there is no polite past tense form of the verb, so でした is added to the negative of the non-past form, 読みませんでした to make a polite past tense. To think of it as the copula here is a mistake. Similarly, でしょう is added to a non-past (or past) tense verb to create a dubitative/tentative form, 読むでしょう, since 読もう, as pointed out, is hortative, 'let's [do]'.
The case of お読みですか is somewhat different. This is the same as お読みになりますか, and one can see that the copula is being substituted for the string になります. Here it is reasonable to treat です as the copula; if one were to force a translation, 'is it your esteemed reading?'
In 私はですね, we have an example of one of the many ways of stating a topic in Japanese. The sense overall is no different from 私は:
私はですね、この本を読みたいんです = 私はこの本を読みたいんです
A literal translation into English of course is unnatural, 'it is me [and]', but that is the basic sense in Japanese.
Further, sentence ending のです means 'it is a case of', 'the explanation for [something in context] is that', or the like. It is also often used when there is a hint of reserve or hesitation, so it is commonly used with ーたい like this, frequently also followed by が・けど:
その映画を見に行きたいんですが I would like to go and see that movie but [would you like to come too?]
私もですよ demonstrates how です, as a copula, can be used as a substitute for a constituent in the same, or a previous sentence:
私もですよ = 私も[読みたいんです]よ 'it is me too [who wants to read it]'
In あの子にじゃなくて、あなたにあげる we have the same usage. Without substitution it would be
あの子にあげないで、あなたにあげる。
English is similar: 'I will give it to you, and not [give it to] her'.
読まないとです makes no sense as it stands. The only way to make sense of this syntax is with some context like this:
お暇があると、それを買ってくれない? 全然急ぎではないんで、お暇があるとですね。
If you have time, could you buy it for me? It's not at all urgent, so, [it's] if you have time, OK?
And here, です is the copula.
The standard reference for anything to do with Japanese grammar is Samuel Martin, A Reference Grammar of Japanese.