Why Does Print Module Export Increase File Size and 'Stroke' Borders

A digital file does not have any dimension in millimeters, so you can't really talk about a 10 mm wide border in a digital file. The resolution in ppi (not dpi) is only relevant when the file is printed. So if you would want to have a 10 mm wide border when printed, then the math would be as follows: Your print will be 300 pixels per inch. An inch is 25.4 mm. So in order to get a 10 mm wide border, that border needs to be 300/25.4*10=118 pixels wide. 

You can't compare the file sizes just like that. When you export your image via the export menu, you will export it at the original size (unless you resize it in the export dialog). When you print to file, you will print a file that has the dimensions of the chosen paper size, multiplied by the resolution in pixels per inch that you chose in the Print module. That could give quite a different size image. It could be smaller than the original, but it could also be bigger.