Is it permissible to perform Tayammum for Fajr prayer due to severe cold and fear of illness? And is it permissible to perform Tayammum with wet dust due to high humidity in winter?

If water can be used, even by heating, then ghusl becomes obligatory, and tayammum is not permissible unless one is certain that water will cause illness, worsen it, or delay recovery. Hot water must be used if possible, or one should wash limb by limb while covering the rest of the body. Scholars have differed on tayammum with damp soil or clay. Abu Hanifa and Malik permitted it, while Al-Shafi'i and Ahmad forbade it. Ibn Qudamah holds that tayammum is not permissible except with pure, dusty soil that adheres to the hand. He mentioned that Malik and Abu Hanifa permitted tayammum with a rock that has no dust, with damp soil, and with everything that is from the earth’s genus, such as quicklime, stones, snow, and gypsum.