How to sum columns and cells with dates and simple criteria?

I do have some advice. It would begin with redesigning your workbook. Seriously. You've approached this in a way that forces YOU to do most of the heavy lifting, using Excel as you would work with paper ledger sheets. For example, in addition to needing to come up with a set of formulas to summarize each month's data by category or key word, it's clear that you have data for each of the last seven fiscal years, each year in its own tab. You could put all of those together in one large dataset (the dates are already year specific, so you don't need to separate them into their own sheets/tabs). 

IF you were to assemble all your data into a single database, then let Excel demonstrate its Pivot Table magic, it would do those month-by-month summaries of hours per column C categories. You wouldn't even need to write a formula. Excel would be doing ALL of the heavy lifting. Here's an introduction to Pivot Tables. There are also quite a few YouTube videos.

I've attached an example of how Pivot Tables can take a set of data with dates and other characteristics and summarize the data by month and category. Your data are comparable.

If this isn't a confidential workbook, and if you are willing to post the actual workbook rather than an image, I'd be happy to modify it and get you started. And, seriously, that would be more helpful to you in the long run than fixing this formula.