Can the Uno handle two serial port functions ?

Using Arduino Networking, Protocols, and Devices
23 
  pinMode (0, INPUT);  // rxd
  pinMode (1, OUTPUT);  // txd

You must not do that. Once you called Serial.begin() pins 0 and 1 are handled by the hardware you must not call any functions that change the hardware configuration.

Or do I need to add some sort of polling code so that data gets echoed back to the serial device also?

If you want to see the characters you type in the hyperterminal, you must echo them back (or activate local echoing in the program).

Can you describe again, what your problem is? Describe what works and what doesn't as well as what you expected.