Is marked as error?
It should still be accepted.
突然 at the begiinning of this particular sentence feels bizarre. I would guess that if the verb is 有 then 突然 cannot be at the beginning.
Yes, it sounds rather unnatural, 今天他突然有…… would be better, although the given sentence is slightly more idiomatic.
No idea. I've reported it.
I see no reason why "他今天突然有重要的會議，所以不可以來。" shouldn't be accepted ("可以" versus "能").
The given sentence is better, 可以 is more like he is physically unable to make it whereas 能 fits better since he could if he were to miss that important meeting, not that it'd be very wise!
Is it incorrect if I start the sentence with 今天突然他...?
今天他突然 and 他今天突然 are both good for my native ears, while 今天突然他 isn't.
I struggle to imagine the context of this English sentence...