Is my transplant ruined with cold water?

There is no correlation between the temperature of water used in washing the transplanted area and outcome. I would not worry about this incident. In the first 3-4 days, care should be taken not to pul out he grafts, which would be marked by significant bleeding should that occur.

Cold water will not damage or harm your transplanted hair.  If you are concerned it is best to contact your doctor.

No there should not be any issues relating to water temperature with the exception of boiling or scalding water. Cold water is fine.

No, temperature of the water should not have a substantial impact. As long as you are following all the other instructions correctly your grafts should be fine.

The cold water will not effect your grafts. Continue to follow your doctors post op instructions and happy growing! 

An FUE is treated just like a regular hair transplant with regard to the recipient area, but the donor area has open wounds which require daily washing with soap and water. Within 3 days of surgery, you can resume full activities, heavy exercises if you wish. The recipient area requires daily washes as well to keep the recipient area free of crusts. I generally recommend the use of a sponge and supply my patient with a surgical sponge to fill with soapy water and press on the recipient area daily. By repeating this daily, all crusts can be washed off without any fear of losing grafts. IF any crust are present, use  a Q tip and dip it into soapy water, and roll it on the crusts and that will lift them off without dislodging them, but never rub them, just roll the Q tip on the recipient crust. I like to see no evidence of any crusting in the recipient area and the crusts from the donor area gone in 7-10 days with daily washing

The temperature of the water will not affect the grafts. Be careful with the direct pressure of the water from the shower on your grafts and also do not rub the area that has been transplanted. After 7-10 days though you do not have to be as cautious around the transplanted grafts.