Yes, you can.
No, you can't.
It depends entirely on the devices you are connecting the TC to. If it's a critical application, I'd suggest using a dual TC, one to each device. If you cannot, or don't want to, you can try connecting up both, as long as you have a way of verifying proper operation.
One big problem with TC's used in parallel is that some signal conditioners actually configure the TC as part of a bridge on the input, in which case, any external connection will cause the readings to be invalid.
The best solution for you overall, may just be to use one TC, and a stand-alone TC to 0-10V or 4-20mA signal conditioning module, and then parallel the outputs of that to your other devices.