Ice (cryotherapy) is a treatment for the inflammation which is secondary to the tendon injury its self. The inflammation is what usually causes the pain and can lead to additional compression injury of the surrounding tissue or chemical irritation of surrounding cells. With all of that said inflammation is a normal part of the healing process and should be controlled, not eliminated.
The way that ice treats inflammation is by numbing pain and constricting blood vessels to reduce the amount of fluid flow to the area thus decreasing the amount of swelling. There are risks to using ice as with any treatment. The risks being that of cold exposure such as frost bite, blisters, pain, etc...
To avoid this risk is not usually a matter of temperature (the goal often being a reduction of 10-15 degrees Celsius) so much as duration of exposure. It is true that the duration will vary based on the temperature, but when using ice in a clinical setting the temperature is usually 0 degrees because the ice is melting. To further reduce that risk it is often recommended that ice directly from a freezer never be placed directly on the skin, but should be wrapped because it could be colder than 0 degrees.
The general rule for cryotherapy is to apply ice until the area is numb or 20 minutes, which ever comes first. There is no definitive research out there regarding cryotherapy, and there are many different modalities and techniques used clinically. There is no target temperature that every clinic uses and measurement would not be cost effective. One target temperature goal cited was 10-15 degrees Celsius of reduced temperature.
"...the studies reviewed were inconsistent in
describing the changes in swelling, blood flow, heart rate, blood pressure,
intraarticular temperature, rheumatoid arthritis, monosynaptic reflex, and the
Additional information regarding magnitude of cooling with different modalities can be found here.