"He suddenly has an important meeting today, so he cannot come."
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.
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!