Firewire can be special icon and color.
Maybe it is just the G-Tech interface that affects the way it is displayed.
If it works and is doing backup and is selected in the control panel (and from menu bar) why worry.
I'd be more worried that Disk Warrior is limited in the size of directories it can deal with unless it has been rewritten to be 64-bit and can handle TimeMachine volume directories that get larger than normal drive directory and as you get into large backup sizes.
I am not sure why the Icon does not change. I just switched over to a 3TB drive in an external enclosure made by MacAlly (by copying over the old stuff and asking Time Machine to take it over the larger drive). I did see the Icon change from the regular orange external FireWire to the Blue/Green Time Machine backup drive icon, but I am running 10.6.8 and it is a different enclosure.
when I click the Time Machine circle icon in dock it does not launch TM window , any idea what's wrong ?
The conventional wisdom when an Application will not launch is to use Disk Utility "Repair Permissions". This should only be done AFTER you have completed the "Repair Disk" step, which it looks like you have already done.
Time Machine is really strange in what it presents when you invoke the Application (to examine your backup or do a Restore). If there is no Finder window open, it appears to do nothing. It is also not quite a real Application -- most of it is in the Operating System, so it may not look at all like a "regular" Application.
I usually invoke Time Machine (either control Panel that runs backup or the Application that runs restore) by using the circular arrow Icon on the MenuBar.