The control you are trying to build is not incredibly complex, but it involves several different methods.
I would suggest creating your own control from scratch, and then inherit the existing one. Using CompositeControl as a base would probably be better as it gives you more flexibility.
For HelpTextTemplate / ContentTemplate you want to create some ITemplate containers, look at this http://msdn.microsoft.com/en-us/library/aa478964.aspx article on how to set this up. Since you can access content / controls in the HelpTextTemplate, check out this article to access them: ASP.Net ITemplate - How to Declare .
For tabs, as this is custom, I would probably avoid AjaxControlToolkit. Instead, I would include a link to the jQuery UI and use the jQuery user interface tabs: http://jqueryui.com/demos/tabs/ . Your CompositeControl just needs to display some divs, ul / li elements, and it will be useful for you to make tabs.
If you are hooked on using the AJAX Control Toolkit, you can still. You need to instantiate the instance in your custom control, add it to the control tree, and then use this technique: http://msdn.microsoft.com/en-us/library/0e39s2ck.aspx to transfer the contents of your template to tab pages.
The ability to drag a control from a toolbar onto your page is simple; if your server management library is already part of the same solution as your website, then it just appears. In the worst case scenario, you can use the "Add Items" option and add the DLL while viewing it. Regarding how a Click event is fired when a button is double-clicked and executed through a class attribute, check out this guide to setting up default events: http://msdn.microsoft.com/en-us/library/43sxkdeb .
As for embedding javascript in the library, these two questions relate to how to do this specifically for the jQuery user interface, if you decide to go a different route, it should still be appropriate: How to embed the jquery library in asp.net user server control ? , http://forums.asp.net/t/1599621.aspx/1 .
Regarding development-time support, try updating a Microsoft article on this subject (including a sample): http://msdn.microsoft.com/en-us/library/aa478960.aspx or this CodeProject article on it: http: // www. codeproject.com/Articles/9227/ASP-NET-Server-Control-Design-Time-Support .