answering in order:
1) deku has an actual costume for when he goes out and does his stunts – he wants to be a recognizable figure after all! – but after becoming a familiar figure around the classroom, you bet he would wear something really bizarre. personally rooting for him to wear slogan shirts that say things like “too cool 4 school” and “you snooze, you lose” except “snooze” has been crossed out and someone has written over it “follow the rules”
2) i believe that deku has an alarming proficiency with any sort of bladed weapon, and that includes axes. skills include juggling and hiding an improbable number of them on his person
3) yes. his clothes are all pockets. reach into your local boy’s pocket and you will find a surprise
4) izuku setting up a murder mystery dinner party for class 1-a and 1-b is something i did not know i needed, and now is going to happen posthaste
5) yes. i love paranatural. with my life