Is it permissible to perform tayammum and ghusl in the aforementioned cases, given the presence of a spiritual illness or a fear of ghusl?

Ghusl (definition) is only obligatory if one is certain of the emission of semen, which necessitates it. If ghusl becomes obligatory, it is not permissible to perform tayammum (definition) when one is able to use water, and it is possible to warm the water when the weather is cold. Tayammum is not sufficient when one is able to use water. Tayammum due to fear of the expiration of the prayer time is permitted by the Malikis but prohibited by the majority of scholars, and the fatwa (definition) is based on the opinion of the majority. If you performed tayammum where it is not prescribed for you, then it is more cautious to repeat the prayers, and this is the opinion of the majority. As for tayammum due to fear of the expiration of the prayer time, you may follow the Maliki school and not repeat those prayers. If you were abandoning ghusl altogether and sufficing with wudu, then all your prayers are invalid and must be repeated according to the majority of scholars.