Is it permissible for someone suffering from asthma and severe allergies to perform dry ablution (tayammum) instead of ritual bathing (ghusl) after a wet dream, fearing for his life, and is it permissible for him to pray at home due to severe cold, fearing illness?

If you wish to pray while in a state of janabah (major ritual impurity), you are obligated to perform ghusl (ritual bath) with water, if you are able to, even if it requires heating the water and bathing in a warm place. However, if you are unable to do so, and you are certain that using water will cause illness, worsen an existing illness, or delay its recovery, then you are permitted to perform tayammum (dry ablution) as Allah Almighty says: "And if you are ill or on a journey or one of you comes from the place of relieving himself or you have touched women and find no water, then perform tayammum with clean earth." If it is possible to wash only part of the body, then according to the madhhab (school of thought) of Al-Shafi'i and Ahmad, one should wash what is possible and perform tayammum for the remainder. Whereas, according to the madhhab of Abu Hanifa and Malik, if most of the body can be washed, then there is no tayammum, but if only a small part can be washed, then one performs tayammum and no ghusl is required. Performing congregational prayer in the mosque is obligatory for an able man, but for one who has a valid excuse that permits absence, such as an illness that makes walking difficult, there is no harm in him praying at home. If your attempt to go to the mosque will certainly or probably result in illness, then pray in your home, and there is no sin upon you.