Dependent combobox to provide matching row count based on 3 criteria

I am looking to count the number of rows that match the criteria that will fall in linked cells for each of 3 comboboxes. I did not want to go the data validation route as I need the drop down to be clearly visible at all times and not when a cell is clicked.

1. How would I dynamically be able to populate the combo boxes based on the previous selection(dependency)?

2. How would I be able to count rows that match criteria on another sheet if 1, 2, or 3 columns match.

I have provided an example spreadsheet. Any help is appreciated. I tried using SUMIFS, SUM(PRODUCT, and COUNTIFS but they did not produce the desired result. Any help is appreciated.