Hi Lars,
For the AjaxToolkit the only other thing I can think of that could be a factor is if your host has a different version of it installed in the GAC on the server and maybe under Integrated it uses the wrong one but under Classic it does not.
I cannot troubleshoot problems in your custom features. The only thing I can suggest is make sute the url mappings are correct under Administration > Advanced Tools > Url Manager.
For Classic you will only be able to use urls that end with .aspx, extensionless urls will not work.
Make sure you are using the Web.config file that shipped with this version of mojoPortal do not try to keep your old one. Copy machine key and any other customizations to the new Web.config
Best,
Joe