Correct!
Correct!
This logic seems strange. "If I see something, I attach something" - that makes no sense. If you read a text, you do not need to attach anything. If you speak, then you have to know which word should be used. So, you have to learn how those words are used.
No, that is not correct, I did not say that "нечего" is always used in genitive! I just explained why genitive is used IN YOUR SPECIFIC EXAMPLE.
"нечему", "нечем", "не о чем" are all valid options, depending on the case you need.
There are 3 distinct things: ничто, нечто and нечего. All of them have different meanings, and are used differently.
1. Ничто is a negative pronoun equal to "nothing". It is pronounced with the second syllable stressed: [ниШТО]. It can be used in any case:
Nominative ничто (Ничто мне не поможет - Nothing will help me)
Accusative ничего, pronounced [ничиВО] (Я ничего не вижу - I see nothing)
Genitive ничего (same as accusative, I cannot think of a suitable example)
Dative ничему (Я не удивляюсь ничему - I do not surprise to anything)
Prepositional ничем (Он мне ничем не помог - He did not help me by anything)
Instrumental ни о чём (Мы разговаривали ни о чём - We talked about nothing)
Note: all the forms have their stress on the last syllable.
2. Нечто is an indefinite pronoun which means "something". It is pronounced with the first syllable stressed: [НЕчта]. There is no negation in this word! Note: ничто and нечто are spelled and pronounced differently.
Нечто can only be used in nominative and accusative:
Nominative нечто (Случилось нечто невероятное - Something impossible happened)
Accusative нечто (Я увидел нечто странное - I saw something strange)
Нечто is a bookish word, not so often used in colloquial speech. You may always replace it with "что-то":
Случилось нечто невероятное = Случилось что-то невероятное.
Я увидел нечто странное = Я увидел что-то странное.
Нечто cannot be used in any other case (except nominative and accusative). If you need a different case, just replace it with "что-то":
Они разговаривали о чём-то интересном. - They were talking about something interesting. (You cannot use "нечто" here, since this case requires prepositional. But you can use "что-то" instead: "о чём-то").
Another example: there is a famous philosophical question
Почему существует нечто, а не ничто? - Why does SOMETHING exist, but not NOTHING?
It shows that "нечто" is opposed to "ничто".
3. Нечего is quite a special word. It is pronounced with the first syllable stressed: [НЕчива]. Note: ничего and нечего are spelled and pronounced differently.
Нечего is used in impersonal constructions only (i.e. in sentences without any subject). It is used when you want to express the idea like "There's nothing to do", "There's nothing to worry about", "There's nothing to read", "There's nothing to drink", "There's nothing to be surprised to" etc. I think you've got it.
It can be used in any case, except nominative and accusative! (And now compare with нечто)
Nominative - does not exist
Accusative - does not exist
Genitive нечего (Мне нечего читать - I have nothing to read)
Dative нечему (Нечему удивляться - There's nothing to be surprised to)
Prepositional нечем (Мне было нечем дышать - There was nothing to breath for me)
Instrumental не о чем (Нам было не о чем разговаривать - We had nothing to talk about)
Note: all the forms have their stress on the first syllable.