How can I find the total engagements (on organic posts, boosted posts and ad campaigns) on your company Facebook page, for a certain period of time?

You can use Facebook insights to see the engagement rate of organic and promoted posts of your company. But if you are talking about competitors or any other company than you have to subscribe for some paid social analytics tools to get data you are looking for. By the way you can't get the complete data of your competitors , the tools will give you limited data which you can use to analyze your position. 

You will need to track Facebook activities via Google analytics. Google tag master gives you the opportunity to be able to track whichever third party platform used in any form of Marcomm.

Facebook does have it on analytics but it is very limited and beneficial when using Facebook for paid-for advertisements

Hey Sarah. I have so far been doing a mix of FB's own analytical tools along with some help from evaluation resources like Quintly to get a grip - albeit tenuous - on the actual yield. Problem: How FB is handling these numbers at the back end is in the gray area as of now. Hopefully the current massive clean-up movement sweeping the digital world will end up shining more light on these critical areas as well.