Cuda 7.0 with Tesla K20 (driver v346 support) and GeForce 8800 (driver v340 support)?

I am on Ubuntu 14.04 LTS.

As I understand to use Cuda with GPGPU like Tesla I need another graphic card as well since Tesla K20c has no display port. If not for it I could not use a monitor and I cannot see anything.

However Cuda 7.0 require driver version 346 or more, when I install Cuda 7.0 it broke my GeForce 8800 GT I am using for monitor connection since it seemed to support only 340 which is for Cuda 6.5 (not an option for me, I need cuDNN v3 support) and cause the login loop problem.

I don’t plan to use the GeForce 8800 for Cuda computation. Can I install the driver for only Tesla but use the old one for display GPU?

No. As long as the 8800 is in the system, the driver you install must be compatible with it.

You can buy a new 730GT for $50 that will be just as fast as 8800GT: