1: Configuration > Index Listing will let you do a lot of things with what shows.
2:You already have the categories appearing on the main page content area, and they work from the DB. The appearance can be altered using the stylesheet and possibly some tweaking in tpl_ and module files, depending on what you want. If you make your own HTML to do this, it will not be dynamic and you will have to adjust it any time you change a category.
3: Are you wanting the big categories spread to appear on all pages? That seems unwise, as they take up a lot of prime real estate. Or what is it that you do want?
PS - If you want the categories not in a sidebar on all pages, use the categories-tabs menu in the header. It will appear below the logo, and can be customized in appearance and even function.