is it possible to make a character portrait appear as a pane beside the map?

Not right now... Yas ......

It shouldn't be difficult if you know HTML/CSS. You can insert bits of javascript in the "UI initialisation script" to rearrange the page layout. Where would you want it? You could chop off some space from one side of the map frame and put an image there; or you could add an extra pane to the sidebar easily enough.

I would recommend putting javascript that creates the extra pane in the UI initialisation script, and then defining a function which you can then use later if the image needs to be changed. You can then set an attribute to the current image filename, and call the JS function from a changescript.

The simplest option would probably be using the custom status pane and putting a picture in it, which doesn't need any advanced scripting.
Or there's nothing to stop you putting an image in a status attribute.

You can do this, but you'll need to add new HTML/CSS/JavaScript elements to the game, depending exactly what you're trying to do.

In fact, Quest allows you to turn off the entire default UI and create a whole new one, if you have the know-how to do so.