For the ask game:
Izuku can see and interact with quirks as entities separate from their hosts.
1 - Izuku has always been able to see quirks, like ghosts haunting their holders, but he has learned to shut up about it because everyone calls him a liar. It's hilarious because quirks have no sense of social boundaries and will casually reveal the deepest secrets ever while Izuku is trying to keep a straight face.
2 - Kacchan is Explosion's nickname and Bakugou is called Katsuki. Bakugou is very confused as to why Izuku is switching between names for seemingly no logical reason.
3 - OFA is full angelic eldritch. Izuku took one look at this quirk and SCREAMED. OFA, very confused that someone could notice him, screamed back. But the good news is that OFA can bring different vestige to the front and Izuku had been acting as an interpreter for Nana to speak with All Might and Gran Torino.
4 - Here is how the conversation between Shouto and Izuku at the Sport Festival happened:
Shouto: [tragic backstory noises]
Half Hot Half Cold: "WHY DOES HE HATE ME SO MUCH? T_T I AM DOING MY BEST TO BE A GOOD QUIRK! T_T HE WOULD PREFER TO GET FROST BITES THAT TO USE ME CORRECTLY! T_T"
HHHC: *full sobbing*
HHHC: *wails of despair*
Shouto: "So, are you All Might's secret lovechild?"
Izuku: "..."
Izuku: "Hey, Todoroki, do you take constructive criticism? BECAUSE HOW ABOUT YOU JUST BE NICE TO YOUR QUIRK???"
5 - "Hey, Aoyama, I don't want to pry but... how come your quirk told me it was kidnapped?"