Without ال it would be wrong in this case. When talking about things in general, using a general statement like that, you need ال. It's like French in that way.
Here are some examples of different phrases:
an important thing (a noun and a adjective, both of them undefined)
الكتابةُ شيءّ مهمٌّ
Writing is an important thing (a nominal sentence, al-kitaabatu (the writing) is the subject (المبتدأ) and shay'un muhimmun (an important thing) is the predicate (الخبر))
Writing is a thing (a nominal sentence, al-kitaabatu is the subject and shay'un is the predicate) (I don't know if this sentence makes sense in Arabic though)
Writing is important (a nominal sentence, al-kitaaabatu is the subject and muhummatun is the predicate, note that muhimmatun is in feminine form, because it functions as an adjective and should therefore follow the noun)
Writing of a thing (pronounciation: kitaabatu shay'in) (an idafa contruction, you can tell that it is an idafa because the first word has no al- in the beginning but ends in ُ and not ٌ and the last words ends in genitive (i or in))
Writing of (an) important (pronounciation: kitaabatu muhimmin) (an idafa construction) (muhimmin functions as a noun here and is is therefore not in feminine case (but could be maybe), many Arabic words can function as noun or adjective depending of the situation, but I am not sure if this particular example makes much sense)
كتابةُ شيءٍ مهمٍّ
Writing of and important thing (an idafa construction) (The word importantant (muhimmin) functions as an adjective to thing (shay'in))
The meaning of a phrase and if it is correct or not depends much of the final vowels. The problem is that the final vowels are not written in most of the questions in Duolingo (maybe reasonable because they are not in most Arabic writing either).
الكتابةُ شيءٌ مهمٌّ = writing is an important thing
is probably a more common phrase than
كتابةُ شيءٍ مهمٍّ = writing of and important thing
but both of them are still correct
If you omit al- then you're referring to what's being written 'Writing ...something...'. But with al- you're referring to the act of writing itself 'Writing is ...'
No, in this sentence you have to use al- otherwise it’s wrong. To refer to a specific written text using this sentence, you’d add هذه in the beginning.
I selected kitaba (without al-), thinking it would mean "the writing of an important thing". Why isn't that accepted?
I know that duolingo didn't teach us this construction yet at this point, but I think it shouldn't tell me I'm wrong if my answer is possible.
"the writing of an important thing" is an idafa construction in Arabic. This is how you would write it: كِتَابَةُ الشَيءِ المُهِمِّ (kitaabatu ash-shay'i al-muhimmi)
كِتَابَةُ (kitaabatu) is the first word of the idafa construction. The first word of the idafa construction never has al before it (but it ends in u or a or i just like words with al before it, and it is considered definite).
الشَيءِ (ash-shay'i) is the second and last word of the idafa construction. If that is definite, it means the whole idafa construction is definite.
المُهِمِّ (al-muhimmi) is an adjective to الشَيءِ
As I wrote in my comment above, if you want to say "writing of an important thing" it would also be an idafa construction and look like this: كِتَابَةُ شَيءٍ مُهِمٍّ