An update on the previous.
Working with another web page (in.yahoo.com) and adding a sleep 1 after the $ie->get($url) solved the problem of setting a value to a field and submitting the form correctly.
However, the last question remains. Can I open a browser window through Win32::IE::Mechanize, then let the user type the input (e.g. some text for search) and grab this input in some way through this Perl module?