Is my prayer valid if drops of prostatic fluid/urine come out after performing istinja' and placing a tissue over the private parts, knowing that I cannot change my clothes or wait? And is it necessary to purify the underwear and trousers if the wet tissue falls and touches them after the prayer?

The prayer of the questioner is valid if he is not certain of the emission of wady after placing the tissue, as the default assumption is that nothing has exited. However, if he is certain of the emission of wady and its discharge does not take up the entire prayer time, then his prayer is invalid and must be repeated, because it is obligatory to wait until it stops, then perform ablution and pray. But if the emission of wady does not cease until the prayer time expires, then his action of protecting himself (from impurities), performing ablution, and praying is valid, because he is considered to have incontinence (salasil). As for the tissue, if he is certain that it became impure and that it touched his trousers while wet with impurity, then the trousers have become impure, and he must pour water over the impure spot. If he doubts whether the impurity touched them, then the trousers are pure.