A gerund is a verb functioning as a noun. In this case, the gerund is "band-aid" and it is the direct object of the sentence, where "on the cut" is the phrase of the indirect object, which is the "cut". Hope this helps!
Answer: Band-aid, functions as the direct object
Third person omniscient would be my best guess. Hope this helps :D <span />
"He exercised while I watch television." because the tenses of the verbs don't match--"exercised" is past tense and "watch" is present.