1: Actual taming wild animals, "Beast Tamer", though wouldn't want it gender locked. So maybe a job rather than a class. Have them tradeable as well. We have "Tamer", but never seen it tame anything. (PSO2 for reference)
2: An underwater dungeon, finding a large door, can breathe on the inside, reach the end and there's a large underwater boss, get special gear from it. I am aware this game doesn't do dungeons, but would've been nice to have at least one.
3: .... I got nothing, hard to think of something BDO doesn't already have... companions? Nah, I get the feeling that with every player using a companion, the game will lag and be unstable. Companions were used in SWTOR, and a game called Onigiri. Warframe uses a pet companion system, specters, and a railjack NPC crew. Oh... maybe a railjack crew would work. Currently in BDO, we can hire and station sailors on the sailboats/frigates, but they don't appear on the ship like the maids do when summoned. So maybe having them appear, man the guns while you focus on sailing might be good, could control when they would be aggressive and hunt sea monsters with them. In warframe, the crew mans the guns and even pilots the ship, while you can run around, and even leave the ship in their complete control. I think BDO could benefit from that greatly, could control where they would pilot by using the map, have them control the guns, and you could just fish the entire time of the trip. Another companion that would be great, is a Shai system. For those who don't know, while you play a different character, you can summon your Shai character which will play a selected song for you. I do this during combat a lot, I have a character who looks like Geralt of Rivia, the witcher. I have the Shai play "Toss a coin to the witcher", during my fights with bosses, bit odd I know, but I have fun doing that since it's a lot like Geralt's adventures with Dandelion. So having a system where you can loop the Shai's music, and have them follow you would be cool honestly.
3 summed up: Companions. I don't expect anyone to read the entire thing. Lol
I don't know, I like BDO a lot, so this is just my opinion of what I could think of right now.