How to invoke testcomplete functions(test cases/scripts) from external application?

It depends on how you want to set this up, but Test Execute is probably your best bet (Licensing and all).  As far as the GUI portion goes, ASP.NET is an easy way to latch on to the TC/TE Integration API and allows you to control and manipulate test execution with C#/VB's built in event handlers.  Keep in mind, however, that you will have to give your "external" application a great deal of permission in order to effectively deploy such a solution. Of course, determination and will could generate a much different deployment model.