Is it permissible to eat with the left hand, and to perform ablution (wudu') by wiping, and tayammum with one hand due to an injury?

If you are excused, there is no blame upon you for eating with your left hand until your excuse is removed. You must wash with your left hand what needs to be washed from the ablution organs, and if a part of your body is sick and washing or wiping with water would harm it, you should perform tayammum (definition) for it. Tayammum is not permissible except when water is unavailable or one is unable to use it. If tayammum is permissible for you, then do what you can, according to the Almighty's saying: "So fear Allah as much as you are able."