This is a very versatile phrase as it is used in numerous situations, such as the beginning of a competition, when you are about to be done a favor, etc. In general, it should be used when you want to express your gratitude or respect towards someone in any situation where they will be doing something for you or on your behalf. There are also different levels to this phrase depending on the level of respect you have to show, which is dependent upon whether someone is your superior or subordinate. Variations of the phrase from casual to most respectful go as follows; よろしく どうぞよろしく よろしくおねがいします どうぞよろしくおねがいします よろしくおねがいいたします どうぞよろしくおねがいいたします There may be even more complex and respectful forms of this phrase, but I haven't learned them yet.

Yes,you are correct. Any of the variations are mainly used for the meanings of "greeting" and "asking for a favor" and might be aggregated into these two.

A lot of different situation you can use. Before start meeting or discussion first day of your part time job first time to join something activity group if you ask somebody, who is master of something, to how to do u go to a mail office and after you finish payment etc....

Would before video games be too formal?

Before start a game such as Chess.