here are my answers:
1) It's possible to add a dropdown to the end of the list that shows other titles, but this would require to add a second loop, that will make your pages heavier to be created. Every loop requires resources: a loop for the page's content, a loop for every widget generated by "Posts in Sidebar" and, adding this feature, will require an additional loop for every widget. Unless I'll find a smart method to avoid a second loop, I do not recommend this. I'll let you know.
2) Do you mean the title of the widget?
3) This is not possible, because the appearance of a website is different from another website. The only choice is a HTML dropdown, like those on your son's website, that can be stylized via CSS.
4) "Posts in Sidebar" has been designed to retrieve posts, not pages. This would require adding a "stranger" (for this plugin) feature... For the same reason explained at point nr. 1, this feature will require an additional loop for every widget. But... why do not use the WordPress "Menu widget" that you already have among other widgets? With this widget you can show all types of links, including categories, posts, pages, or external links. The downside is that you have to manually choose what to display. Anyway I'll take a look if it's possible without affecting your site's performance.