Is dedicating siesta and sleep time to supplication (du'a), especially supplicating for matters such as not having a wet dream or scratching a rash on the body to avoid obsessive thoughts, considered an innovation (bid'ah)?

There is no harm in you reciting these supplications, for they are part of adopting a regular devotional practice and are not an innovation (bid'ah). Many of the Salaf (early Muslims) used to regularly recite supplications they needed. However, you are not permitted to claim that they are a prescribed Sunnah for everyone, or to believe that there is a special virtue in consistently reciting them.