- I would like to know if Noun+でいる is grammatically correct.
- Is "つもりでいる" an exception?
Yes, it is grammatically correct, and "つもりでいる" is not an exception.
First of all I highly evaluate the question by the questioner.
The reason for this is that the expression "つもりでいる" in Japanese is quite familiar, and it is easy to understand that "つもり" is a noun, but if you replace "つもり" part with another noun at hand it generally does not make sense as Japanese. Why? It's very interesting to see what kind of secret is hidden in "つもりでいる".
I searched for the example of "***でいる" in BCCWJ: Balanced Corpus of Contemporary Written Japanese here. As a search result of 500 examples, most of the examples having a noun that corresponds to "つもり" part was "つもりでいる" posted by the questioner. Though other examples were not so many, I could collect some, so after classifying the examples, I'll show them as follows:
(1) 一人{ひとり}でいる、二人{ふたり}でいる、 皆{みんな}でいる
(2) 正直{しょうじき}でいる、 嘘{うそ}つきでいる
(3) 笑顔{えがお}でいる、 泣{な}き顔{がお}でいる、しかめ面{つら}でいる、しかめっ面{つら}でいる、 知{し}らん顔{かお} でいる
(4) 気持{きも}ちでいる
(5) ありのままの姿{すがた}でいる、ずぶ濡{ぬ}れでいる
(6) 元気{げんき}でいる、達者{たっしゃ}でいる、病弱{びょうじゃく}でいる
(7) 友達{ともだち}でいる (、母{はは}でいる)
(8) 孤独{こどく}でいる
Next, I highly appreciate user 4092's finding and/or knowledge that "つもりでいる" could be explained clearly by letting it compared with and clearing the difference with "つもりである".
With this knowledge, it became also very easy for me to explain how "つもりでいる" works.
I know my understanding partially overlaps with user 4092's finding and/or knowledge, I think the biggest difference between "である" and "でいる" is that the former describes the objective fact, state or condition, while the latter explains the fact, state or condition combined with the will or intention of the person who has them relating to them. In short, it is the difference between objective state and subjective state.
As for "つもりでいる" that the questioner posted and there were the most usage examples, it could be translated into English as "I have an intention that", "I'm trying to" or "I'm going to". Unlike the examples shown from (1) to (8), this is a special example in which the meaning of the speaker's will and intention is included from the beginning in the noun "つもり" itself, so when it is used in "つもりである", it seems to be an example where the meaning does not change much as compared with other examples. Nouns of this kind having the nuance of "will", "intention" or "judgement" as "思い", "考え", "判断" and "理解" could be expressed as follows, they make sense and their meanings would not change much used in "である" form as: "思いでいる" - "思いである", "考えでいる" - "考えである", "判断でいる" - "判断である" and "理解でいる" - "理解である".
If you change the examples from (1) to (8) to "である" form, there are many cases sound strange as Japanese. In those cases, if you change them to "です" form, they make sense as ordinary Japanese.
The humble form "おります" of "いる" is often used in a letter, postal card or e-mail in the form of "でおります" instead of "でいる". This expression is conveniently used when expressing the feelings of the sender along with the state of the sender of the letter.
"である" or "です" is a copula and there is no intention or will in its use, while "でいる" needs the intention or will of the owner of the noun, so nouns which have neither will nor intention, such as flowers, trees, stones, etc. that are nouns at hand could not be used in "でいる" form.