All of the Comic References in Wayne Family Adventures #7
Starting Their Own Superhero Team:
Dick co-founded and led two versions of the Teen Titans (the 'Fab Five' Silver Age Titans and the New Teen Titans); he also ran the Outsiders (Judd Winick's 2003 Outsiders run) and two iterations of the Justice League (JLA: Obsidian Age, as Nightwing, and the post-Cry for Justice League, as Batman)
Jason started the Outlaws (New 52/Rebirth Red Hood and the Outlaws)
Tim co-founded Young Justice (YJ 1998 and YJ 2019) and led his generation of Teen Titans (Teen Titans Vol. 3, 2003-2011, and Vol. 4, 2011-2014)
Damian was part of a new lineup of Teen Titans (Teen Titans Rebirth, 2016-2021) & his Super Sons team-up with Jon Kent also probably counts (Super Sons saga)
Changing their Hero Names:
Dick became Nightwing in Tales of the Teen Titans #44. Not included on this list are his time as Batman (Batman: Prodigal and Battle for the Cowl #3/Batman Reborn era) and Agent 37 (Grayson, 2014).
Jason became Red Hood in Batman: Under the Red Hood (2005-2006).
Tim technically became Red Robin in Robin (1993) #181, but was only named as such in Red Robin #1; Drake was in YJ 2019.
Cass became Batgirl at the end of the No Man's Land arc and properly debuted as Batgirl in Batgirl (2000) #1. She took up Black Bat in Batman Inc. Vol. 1 (2011) #6, and Orphan has been her sole hero name thus far in the post-Flashpoint universe, starting with Batman and Robin Eternal (2015).
For Damian and Cass, it's their entire backstories, so pretty much anything touching on that will cover it
Tim was trained by Lady Shiva in his first miniseries (Robin Vol. 1, 1991), published before his solo run in 1993; he also got some training by the League of Assassins during his 2009 Red Robin run
Jason was trained by Talia al Ghul, the All-Caste, and LoA in between being resurrected and his return to Gotham (Red Hood: The Lost Days and the New 52 RHATO run)
The Dick-Deathstroke mentions are either a reference to the Apprentice arc in the 2003 Teen Titans cartoon or Nightwing: Renegade, where Dick went undercover as 'Renegade' and ended up training Slade's daughter, Rose Wilson, in exchange for Slade agreeing to stay out of Bludhaven. Neither one would give him a legitimate bingo (TT because it's not comics canon, Renegade because he was doing the training, not being trained). HOWEVER, Dick should still be raising his hand because of his time at Shrike's Vengeance Academy in Robin: Year One.
Murders and Resurrections:
Jason was beaten and murdered by the Joker in Death in the Family. Technically, his resurrection happened in Infinite Crisis/Batman Annual #25, but really...just read Batman: Under the Red Hood (2005-2006)
Damian was killed by The Heretic, his adult clone, on his mother's orders in Batman Inc. Vol. 2 #8 (2012) and resurrected via a chaos shard on Apokolips in the Robin Rises arc of Batman & Robin Vol. 2 (2011-2015).
Cass died and was resurrected twice during her Batgirl solo series; the first time in a fight with her mother, Lady Shiva, in Batgirl (2000) #25-26, and the second time in a fight with her LoA "adoptive brother," Mad Dog, in Batgirl (2000) #71-73. On both occasions, she's resurrected by Shiva (the first time, Shiva restarted her heart via some mystical thing that's never really explained; the second time, she dunked Cass in a Lazarus Pit).
Dick's wishy-washy murder/resurrection and the Crime Syndicate stuff was Forever Evil (2014). Dick was strapped to a bomb that only worked as long as his heart was beating, so Lex Luthor killed him via suffocation, disconnected him from the bomb, and then restarted his heart. Dick was "medically" dead for less than a minute and was promptly resuscitated, resulting in the ongoing fandom controversy on whether or not his death "counts" for the purposes of the "Dead Robins Club" (lampshaded by Jason and Tim's argument)
Sidenote: even though Steph's death in War Games was later retconned as Leslie Thompkins faking her death and smuggling her out of the country/into witness protection, Steph genuinely flatlined and it was intended to be a genuine and permanent death at the time of publication. If Steph were in the Cave, I assume she would also raise her hand.