I’m referring to the DoJG books for these three grammar points and have gotten a bit confused. Under に関して’s page (intermediate pg 252), the difference between に関して and the other two grammar points is stated as a formality and prenominal form difference (i.e. the need to put to の after について to form についての) for について ,and with のこと, it is stated as a formality difference and also that のこと doesn’t have a prenominal form.
However on について’s page (intermediate pg 280) under related expressions, the dictionary states that the difference between “Nについて” and “Nのこと” is that the former means ’about / concerning N’, but the latter, ‘things about N’. And the following sentences are given for comparison.
[1] A：田中さんを知っていますか。Do you know Mr. Tanaka？
B：ええ、昔から。Yes, I’ve known him for many years.
[2] A：田中さんについて知っていますか。Do you know about Mr. Tanaka？
B：ええ、本で読んで、知っています。Yes, I know of him through books.
[3] A：田中さんのことを知っていますか。Do you know（things）about Mr. Tanaka？
B：何かあったんですか。Did something happen to him？
The sentences are then explained: “[1A] is a straightforward sentence of ‘to know something '. In other words, [1A] expresses direct knowledge. [2A] has ”Nについて” and expresses more indirect knowledge; [3A] concerns knowledge about circumstances surrounding N（i.e., Tanaka).”
Surely this would mean that either the difference between に関して and のこと is more than just a formality and prenominal form difference or that the difference between に関して and について is more than just a formality and prenominal form difference wouldn’t it?
I don’t have nor have ever used DoJG books so I can’t check the contents, but about the grammar points themselves, if you are going に関して x について, there is surely a formality difference. For example, writing 「に関する研究」or 「に関する論文」for “research/paper on xxx” is very common, while writing「についての研究」 sounds a bit childish or poor vocabulary at best. The latter might be used in casual talk, but it would surely be avoided on a speech, formal presentation and such.
About のこと, there is surely a meaning difference from the previous two. (while being on the informal side of the scale, too)
Following your example sentences, if you ask someone 田中さんについて知っていますか it feels like you are asking about basic info (place of birth, work, education, etc) about 田中さん. If they are a famous person you might be confirming the listener’s knowledge before asking about their career, for example. If 田中さん is not a famous person, you most likely just met them but you are curious about something specific and want to check it, such as you feel like maybe you’ve studied on the same junior high school, or that they are the ex-spouse of someone you know. So you are confirming if your listener has info you can use to check what you think.
If both you and your listener know 田中さん, asking 田中さんのこと知っていますか feels like “did you hear about what happened to them?”, hence the book’s example answer. Maybe they got involved in an accident, got divorced or screwed up big at work, who knows.
Comparing に関して and のこと directly feels a bit awkward, because they are different both formality and meaning wise while also having different grammatical uses (you can’t have a prenominal form of のこと, you can’t use it as a sentence connector, etc).