Programming/running a NXT brick with a dead screen, via PC?
There is a tool called NeXTScreen that is available here that will let you see the screen (and remotely control the NXT). There are similar tools for OS X and Linux on the same page.

Once you have this setup, you should be able to use any programming environment that you like. The screen viewing programs should be able to run at the same time as anything else.

For example, you could use the screen viewing program to pair via Bluetooth, then program the NXT using NXT-G and use use the "run" button in NXT-G to start the program remotely.