That is everybody's/everyone's problem should both be accepted
shouldn't it be 문제이다?
문제이다 -> 문제다 (for noun ending in vowel only: this loss of "이" is called 'particle elision' which happens in casual Korean language - not academic works).
can you use 모두 다 in place of 모두의 and still get the same meaning?
모두다의 , yes. 다 is just an added emphasis.
Note: without the possessive 의, 모두다 문제이다 = everyone/everything is a problem.
I don't think so, 다 is the adverb version of 모두. They both mean the same but are not used together.
Are you sure about that? I've seen term 《모두 다》or even《모두다》on few instances before. I also checked it in a dictionary, and this term was listed there as 모두 with increased emphasis.
I can't believe they don't accept that problem is everyone's... I was so confused
It didn't accept "That problem is everyone's." Though the meaning in English between that and "That is everyone's problem." are identical.