Support paste of images from clipboard

## Problem

Clipboard paste support of images in the chat input is not supported.

## Expected behavior

It would save me from having to save a screenshot to a file and then attach it to the chat message. Some AI assistants support this usability feature.

Start by locating the chat input implementation and its existing clipboard or attachment handling. Trace how pasted content is processed, then verify that an image pasted from the clipboard can be attached to a chat message without first being saved to a file.

Written by the indexing model from the issue text.