What are the proper usages and the meaning of くはある / くはない?

Question

I recently learnt the contrastive usages of the は particle and I read the Tofogu article which has an interesting section which says I can use は contrast marker with the adjectives/nouns positive and negative form. But the article is not detailed enough for me to understand exactly what's going on. Here are some example:

い-adjective:

positive form pattern: く-form of い-adjective + は + ある

難{むずか}しくはあるけど、おもしろい。

negative form pattern: く-form of い-adjective + は + ない / ありません

難{むずか}しくはないけど、ただ時間{じかん}がかかる。

な-adjective/noun:

negative form pattern: noun + では + ない / ありません

有名{ゆうめい}ではないが、人気{にんき}はある。

My first questions are:

1. What is the positive form in the case of な-adjective/noun? Something like that: noun + は + ある?
2. Can the word with くはある / くはない only be in the end of the sentence/subsentence? So is the following sentence incorrect?:

可愛くはない猫です。

Currently I interpret this は usage in the following way based on these examples and the article:

The は particle (in these cases) expresses: the statement does not 100% cover the situation/speaker's opinion because there is something else too. The difference between positive and the negative form is the following:

• When the positive form (くはある) has been used then we express the statement covers between 50 and 100% the situation/speaker's opinion.
• When the negative form (くはない) has been used then we express the statement covers between 0 and 50% the situation/speaker's opinion.

So my next question:

3. Is my interpretation correct?

Because I'm not sure. Here one answer under the Why does 悪く(は)ない give a condescending feeling? question says:

When native speakers such as myself say 「悪く(は)ない。」, they often imply that the object of the discussion is not too good. ... English-speakers seemed to often say "not bad" in situations where they clearly meant to say "pretty good". I am not saying that this usage of 「悪く(は)ない」 never occurs in Japanese, but I can assure you that it is rarer in Japanese.

Which is contrary to my conclusion. Because the common interpretation of「悪く(は)ない」is mostly (50%-100%) bad and not lessly (0%-50%) bad. So:

4. Is the「悪く(は)ない」 an exception when we interpret くはない? If not then what is the differences between くはある and くはない?

Answer 1

1. What is the positive form in the case of な-adjective/noun?

ではある. And when do you see it used? When there's a need to foreground a sense of contrast.

彼は確かに利口ではあるが、冷淡でもある。(source)
He is clever, no doubt, but he is cold, too.

本日台風ではありますが営業しております。(source)
Today there is indeed typhoon but our shop is still open.

Of course you can leave out the second half of the sentence while still keeping the sense of contrast:

本日台風ではありますが。

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2. Can the word with くはある / くはない only be in the end of the sentence/subsentence?

Allow me to rephrase your question for you. You are essentially asking: can は be inserted in relative clauses? Namely, you have seen は used in predicative phrases, and you wonder if it can be inserted in an attributive phrases, right?

So in such a sentence, the contrastive は occurs predicatively:

あの問題は難しくはなかった

And you are fine with it. Now you are wondering what happens in an attributive phrase modifying a noun in a relative clause. The answer is affirmative, too. Yes, it does occur.

難しくはない問題だが「直線に関する対称点」の処理ができたかどうかがカギ。(source)
It's not a difficult question, but whether the student can figure out the point of symmetry relative to the line is key.

You probably have read/been told that は can't be used in relative clauses. For examples two answers (this and this) from two of the most prolific and knowledgeable contributors both tell you not to use は in relative clauses. But there's an exception: when it's the contrastive は as opposed to the topic marking は. This is actually explained in that Tofugu article you cite, and I'd like you to take a look at this part:

は IN RELATIVE CLAUSES

Since it's also possible to use の in relative clauses, there are three possible ways to render a sentence like "I was accepted by the school that Jenny failed to get into."

ジェニーの落ちた学校に私は受かった。
ジェニーが落ちた学校に私は受かった。
ジェニーは落ちた学校に私は受かった。

However, it's worth noting that は can only be used in this way if there is some kind of contrast implied, like in the sentences above. It the relative clause is simply describing the noun, with no nuance of contrast, only の and が can be used:

ジェニーの落ちた学校に私も落ちた。
ジェニーが落ちた学校に私も落ちた。
❌ ジェニーは落ちた学校に私も落ちた。

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3. Is my interpretation correct?

No, not really. There's no such difference between affirmative and negative phrases. Again, the key is contrast. When は is used this way, the speaker/writer is trying to show contrast.

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4. Is the「悪く(は)ない」 an exception when we interpret くはない? If not then what is the differences between くはある and くはない?

No, I don't think so.