Is there a way to have the captcha field display a required field message when it's overlooked/hidden, caused by for instance a resized browser window? Now, only a message appears when the captha field doesn't match the given image code.

No there is no way to have the captcha control participate in the client side validation at this time. The captcha control is from the DNN WebControls library. We can look into it for a future version. You can add a label as the last row in the form advising users of the need to fill this out.