Is there a way I can make it so people can't approve their own signups...I want to make it so only officers can approve calendar signups.

Head to Admin > Feature Access, and make sure "Schedule" is set to "officers." Right now it's set to "initiate" which means anyone ranked initiate can create official events, delete them, and approve members.

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It's all in the reflexes.

Ok, maybe I need to do a feature request....would like to have people be able to make their own events and approve people that sign up, but don't want people to be able to approve themselves for other peoples events....We have a lot of people approving themselves for our Kara runs and we end up with drama because 12 sign up and we have to wait list the extra 2 based off of the time stamps....Would like to prevent this from even happening.

Look at the "myschedule" security option in Admin > Feature Access. This does exactly as you're requesting.

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It's all in the reflexes.