PSoC Programmer can be used on its own,and infact Designer(and Creator) simply use the COM object from Programmer.
ICE Cube(havent used it myself,but..) should work with PSoC Programmer too.Atleast for programming.
Raise a case with tech support ..
You should always power the ICE-CUBE to let it operate. The checkbox "external power" is referring to whether the connected POD is externally powered (from the circuit) or if it should be powered by ICE-CUBE
PSoC1 could only be debugged by use of a POD or OCD part (on-chip-debugger). If you wire your PSoC using the 5 ISSP lines, you will only be able to program the part, not debug it.