Using two separate USB connectors, one for power and one for data: do I tie their grounds?
If the grounds of two parts are not connected, the signals from one part will be misinterpreted by the other. You must either connect the grounds or use some sort of isolation circuit between the IC connected to the PC and other components.

I had a project where I had to do something similar. There I powered a device from external power source and used USB cable with V+ wire cut with no unwanted effects and I am pretty sure you can do something similar in your project.