Does the discharge, mixed with a brownish color and an odor, originating from the anus and clinging to the underwear, invalidate the prayer? And what is the ruling concerning past prayers performed in this state?

Slight traces of impurity and its smell in underwear due to sweat are excused. This is the view of the Hanafis and the choice of many verifying scholars. The reason for this excuse is the difficulty of guarding against it, by analogy with flies that carry slight impurities.

Also, the Shari'ah permitted Istinja' (cleansing after defecation) with stones, and it is known that stones do not remove impurity completely, and a slight trace remains which is excused. The Companions used to perform Istijmar (cleansing with stones) and would sweat, and it was not reported from them that they were obliged to purify what stained their clothes from this sweat. So if the area sweats after Istijmar and it touches the clothing, it does not matter, and it is excused. This is due to the widespread nature of the problem and to prevent people from falling into hardship.

Therefore, the slight trace resulting from sweating is excused, and the prayer is valid, while ensuring thorough cleansing of the area during Istinja'.