Spillways

how many fish are swept into the river, when the dam gates are open?
or
do the fish stay away from the current

You asking about fish passing under the gates at the dam that when lifted release their flows from very deep reservoir waters or per your thread title, the overflow spillways at the dams that only flow at the water's surface very rarely and if so very briefly when a lake exceeds capacity?

yes
through the gates

Quite a few Stripers and Whites go over. There were copious amounts taken behind Morris Shepard Dam and Dennison dam when they were running strong last year. I'm not sure about other species, but I know these two come over a lot.