Hello,Daria! I'm Akitaro from Japan. We usually use "選挙" as noun. For example, 上院議員選挙 = Upper house election. "選出" usually means "to be elected" or "elect". In other words,It is used as a verb. For example,彼は議長に選出された。= He was elected to the chairman. And "公選" means "election of public officials by general voters" "公" means public and "選" means selection(select) or election(elect). In my thought,these are same as Chinese. I hope these will help you learn Japanese.

In Chinese, “選挙 (选举)” means election, while “選出 (选出)” means elect or select or choose (a verb). Let's wait for a Japanese answering this question.