Is it permissible to perform tayammum instead of ghusl due to illness, while being able to perform wudu', and is it obligatory to perform wudu' with every prayer after tayammum, or is tayammum alone sufficient?

If there is a concern about using water due to a cold, it is necessary to heat it and purify oneself with it. If the inability persists after heating, then Tayammum is permissible. If it is possible to heat the water or use it in a way that ensures no harm, then that is obligatory. If one is unable to wash all limbs, Tayammum is sufficient. However, if one is able to wash some of them, then one should wash what is possible and perform Tayammum for the rest.