Hello,

Questions for part A:
The search element in the Header builder has the option to show the list of categories existing on the site. This list cannot be edited. Select Categories will shall all the categories.

Questions for part C
You can rewrite all theme and plugin texts via PO file in WordPress. Here is a video tutorial that should help you rewrite your website texts with a Loco Translate plugin https://www.youtube.com/watch?v=D3NsDdMzsls&list=PLMw6W4rAaOgKKv0oexGHzpWBg1imvrval&index=3

Questions for part D
This list cannot be edited. Shop title shows all the existing categories.

Questions for part E
Enter the shop page and change the image on the title background in Image for page title – Upload and set an image on the title background; https://xtemos.com/docs/woodmart/options-for-pages/

Questions for part F
Please navigate to Woodmart > Header builder http://prntscr.com/iyd2pe

Choose the proper header type (your current) http://prntscr.com/iyd333

Delete the text block in the right bottom corner, check before deleting: http://prntscr.com/iyd3h3

Questions for part G
Woocommerce widget allows hiding empty categories only. It also shows the existing categories.
You can create a menu in Appearance > Menu and insert by means of menu widget, however products would not upload by AJAX, you will be redirected to the page.

Best Regards