I need a specific way to show the posts in the grid.
* CustomField1, CustomField2
* below that goes the title which needs characters before and after (quotation marks)
* I need the featured image to show below with specific width and position
* CustomField3 with specific position
NOW: Is it possible to edit the layout via a .php file, so that the masonry function remains, but I can order all parts of the posts to my liking?
Ya it can be done. You can copy the masonry layout to wordpress theme using Copy to wordpress theme link. From there you will get the new moved location. You can locate layout.php file and edit as per your needs.