Is it permissible for a woman suffering from rheumatism, whose pain increases upon using water for ablution (wudu') or ritual bathing (ghusl), to perform dry ablution (tayammum)? And is it also permissible for her to perform tayammum if her ablution is constantly nullified due to the emission of gas?

If purification with water for major or minor ritual impurity is difficult, then tayammum (dry ablution) is permissible. If a person is able to purify some parts of their body with water, they must do so, and then perform tayammum for the rest. As for continuous flatulence, the ruling for someone in this state is similar to that of a person with urinary incontinence: they should perform ablution for the prayer after its time has begun and pray the obligatory and supererogatory prayers. If the emission of gas stops for a period sufficient for prayer, then the prayer should be delayed until that time. If one fears that the prayer time will expire, they must pray even if they pass wind during the prayer or during purification, and their prayer is valid.