Why do I have to click show score for each game after simming a week? Also, is there a way to stop from advancing the drop down to the next week after simming?

In settings you can disable hide scores. Thats mainly for players (an multiplayer leagues) who want to sim a week and then watch games without knowing the scores.

No workaround for the dropdown, but you can switch weeks with the mousewheel right when its done simming to jump back.