Does the contamination of the inner garment affect the outer garment and invalidate the prayer? What is the ruling on prayer when one is unable to change the contaminated garment outside the home? And what is the ruling on reciting the Quran in such a situation?

If a person experiences drops of urine and doubts whether the impurity has transferred from their inner garment to their outer one, the default assumption is that it has not transferred unless there is certainty.

It is sufficient for him to change his inner garment or wash the impure spot; he is not obliged to change his outer garment unless he is certain that it has been afflicted with impurity.

If he does not change the impure garment, he must wash the impure spot. If he is unable to do so, he should pray in his current state. The Maliki view is that he is not obliged to remove the impurity.

Reciting the Quran is permissible for someone whose body or garment is impure, but it is disliked (makrooh).