Checkbox function to automatically add to total

There is no native checkbox function, so I would use Data Validation, in combination with COUNTIF (or SUMIF, depending on what you want to count). Check this example sheet.

To add Data Validation, first select the cells you want to add it to.

Go to Validation in the Data menu

Select List of items in the Criteria-drop down and write "Yes, No" in the textbox. Make sure that Display in-cell button to show list is checked.

Bellow you can see what is shown when you click a cell with Data Validation. You can choose to only enter "Yes" in the Data Validation textbox, if you only want to have Yes as an option, and then delete the content of the cell when necessary. The No-option is only to make it easier to remove a row from the count.

If you really want a check-box you could check this answer. You can use images in combination with scripts to create check-box-like things.

I think the problem is that you're trying to use a spreadsheet like a database, or an application. Why not use something like Ragic instead? You can add different types of fields, including selection checkboxes like you want.

Disclaimer: I work at Ragic and have been designing many database applications with it.