Is it obligatory to inspect one's undergarments after arousal to ascertain if anything has been discharged, and what is the ruling on sprinkling one's trousers with water if madhy (pre-seminal fluid) is found on the undergarments, and how can I ascertain if the madhy has reached them?

There is no need to inspect or examine clothing when in doubt about the emission of madhi (pre-seminal fluid), even if the likelihood of its emission is strong, because the default assumption is its non-emission. However, if you are certain of its emission, then you must purify your body and clothing from it and perform ablution (wudu) for prayer. If you doubt whether madhi has transferred from your inner garment to your outer garment, you are not required to purify what you are doubtful about, because the certainty is that it has not transferred. Impurity is not determined merely by doubt; rather, certainty comes from seeing the trace of madhi on the outer garment.