For every part we create, we use the Custom properties to define PART NUMBER and DESCRIPTION. 
Then inside a drawing you can use the TABLE BOM and select CUSTOM PROPERTY > PART NUMBER
Now if I use a Tool box part they have a field PART NUMBER, for example "11127"
But it is a "SolidWorks" PART NUMBER not a "user" CUSTOM PROPERTY > PART NUMBER
So it looks like the file name is the "PART NUMBER" well tell me who uses just a "PART NUMBER" in a file name.
Is that what I need to do? Doubt we will be able to.
So, you would think that this high tech software would be able to have a PART NUMBER across part, drawings, and add in components that was all linked. Someone please explain why not? or better how to do it? I do even recall out of the box there was a "Part No." and I had to make PART NUMBER to just get the part connected to the drawing. This is the most basic feature I think. Frustrated. Educate me, please.
Just out of curiosity, did you have formal SW training or were you sat down in front of a computer and told to learn it yourself?
Out of the box, part number is pulled from Bill of Materials Options field in the configuration property manager. There are three selections for how SW will populate the field: Document (file) name, Configuration name or User Specified Name. I do not use Toolbox so I do not know how the properties are set.
From your screen shots, I see you have modified the BOM table to use your part number custom property. I will assume Toolbox does not create either a part number or description custom property (although I see you fill in your data on the custom tab; maybe your toolbox parts fills the information in the configuration specific tab).
Fortunately or Unfortunately SolidWorks is customize-able and in many cases can be made to work with your company's workflow. You need to pick which works best for you and make sure all parts are created following that procedure. Since I do not use Toolbox, I do not know if it can be customized to create PART NUMBER and Description field.
Where I work we use configurations, so our custom properties are put in the configuration specific tab. We also have a macro (now an add-in in PDM) that we used long before SW had the Custom Properties tab and other functionality. This "macro" automatically changed the BOM Options to User Specified and populated it with the data found in our part number custom property (we use the part number custom property to populate our drawing format). This works for us but it probably will not work for you since you have probably have many parts done using the custom tab.