Shoutbox

1, 3. Not possible without significant changes in the Ink feature itself, since it's the Ink redistributable that allows you to use Handwriting in the first place. Ink comes with Windows Journal Viewer.
Think of Ink like an embeddable control, much like a button, a checkbox, or even IE (MP!L's Log Viewer and SPNG's dialogs are examples of an embedded IE control). To extend the functions of Ink, it'd need to expose methods of extension, which I doubt it does. The other option is changing Ink itself, and only MS can do that.

2. Ink is used for 'vector image'. Basically, it records lines and colours, not pixel data. A screenshot is a 'raster image', which contains pixel data instead of lines and colours. Converting a raster image to a vector image is an expensive operation that results is a ballooning filesize.
So it's pointless pasting a screenshot as an Ink message, sadly. However, it should be possible to create a Screenshot Sender-alike in script.