How can one deal with the obsessive thoughts of passing wind during ablution (Wudu') and prayer (Salat), knowing that the wind is expelled as a result of attempting to push it out, and is not a natural incontinence? And what is the ruling on using soap to remove a greasy substance from the feet during ablution and then completing the ablution thereafter?

This feeling might be from the whispers of Satan, for certainty is not removed by doubt. You are not obligated by the doubt of passing wind, and you should ignore such doubts.

However, if you are certain that wind exits continuously, preventing you from performing prayer with a valid ablution, then you are in the ruling of one afflicted with chronic wind ( سلس الريح ). It suffices for you to perform ablution for the time of each prayer, and you may pray as much as you wish throughout that time, so long as your ablution is not invalidated by anything else. This is the opinion of the majority of scholars.

For the one afflicted with obsessive doubts ( الموسوس ), it is permissible to adopt the view of the Malikis, which states that a continuous minor impurity ( الحدث الدائم ) does not invalidate ablution, if there is a need for it.