"She is sick. She shouldn't go to work."
yes and no. 去上班 feels to me like go to work on that day, like 我今天去上班 I am going to work today, while 上班 isolated feels to me like a more general statement, like 周末我不上班 I don't work on weekends
Agree. The hover over tips say '上班'. So we would expect that to be accepted.
生病 acts as a verb here, so instead of "she is getting sick" (她生病）we say "she has gotten sick" (她生病了）。
yes, 病 by itself doesnt mean sick as in illness but more so, "sick in the head"