どっと does have meaning of suddenness, but usage of どっと is limited to those 3 situation, according to Digital Daijirin.
Lots of people letting out their voice at the same time.
Lots of people / things coming at one time.
Becoming (seriously) ill in a short time.
We use どっと like this:
観客かんきゃくがどっと笑わらいだす (The audience bursts into laughter)
人々がどっと押おし寄よせる (The crowd surges forward)
どっと病やまいが重おもくなる (Become seriously ill suddenly.) (sentence from the reference)
突然 (とつぜん) is more common, because its meaning is broader than that of どっと.