Is it possible to customise the edit dialogue window that is opened when editing a web part? Customised with regards to appearance, what fields are exposed etc.
yes, of course, but it needs development. It is called a Toolpart and is implemented via EditorPart class. Examples are everywhere just search for it. E.g. http://blog.concurrency.com/sharepoint/create-a-custom-web-part-for-sharepoint-2010/
In terms of properties, those are in direct relation with your what your WebPart class properties exposes. Some are OOTB provided by the Web Part framework but you can choose to extend them.
Let me know if you need more details, see also here http://msdn.microsoft.com/en-us/library/ff597540.aspx C:\Marius
Not all properties are implemented via EditorPart, just drop downs, checkboxes, and custom EditorParts. The standard toolbox inherits from WebPart. Check out this link for implementation:
This is the walkthrough I used to create my first Visual Web Part. You'll use some code like:
You can hide specific settings from the edit window by setting the "WebBrowsable" property to false.