Make GIMP 2.8 open in a maximised window and add an "Edit with GIMP" right-click context menu

According to the person who actually compiles GIMP on Windows, the problem is related to the GTK+ toolkit. And this makes sense, because GIMP isn't the only linux port to suffer from this annoying problem.

Updating to GIMP 2.8.4 (or higher) will resolve the issue in Single-Window Mode.

Otherwise, you will have to find a third-party program that can automatically maximize windows for you. I am not aware of any free ones but Actual Window Manager will do the job. If it's a bit too pricey, you may be able to get the same functionality from some of their cheaper programs such as Actual Window Guard or Actual Window Minimizer.

In regards to your second question, you can use the following registry script to add the 'Edit with GIMP' option to system image types assuming you have GIMP 2.8 installed to the default directory:

Windows Registry Editor Version 5.00

[HKEY_CLASSES_ROOT\SystemFileAssociations\image\shell\Edit with GIMP]

[HKEY_CLASSES_ROOT\SystemFileAssociations\image\shell\Edit with GIMP\command]
@="\"C:\\Program Files\\GIMP 2\\bin\\gimp-2.8.exe\" \"%1\""

To maximize GIMP once it has loaded, you can use AutoHotkey as demonstrated in my answer here: https://superuser.com/a/430863/100787. From there, you can download my Maximize executable or compile the script yourself which can then be called from the command line, shortcut, or batch script.

Example:

Maximize "GNU Image Manipulation Program" "%ProgramFiles%\GIMP 2\bin\gimp-2.8.exe"