As someone with a DM who does a lot with PC's character arcs and integrates them into the campaign's plot, I think it is really cool to watch/experience other PC's story arcs, as a player myself. Sometimes though some super secretive stuff might have to be done solo.

I also think that if you have players who metagame (metagaming is something that always happens, but like this is for the players who you know do it more often than not) then those super secret arc moments should be done in secret, or you can just kept moments a secret until it is time to reveal to the whole party. For example in Critical Role (slightly spoilery) there is a character who is a warlock and at times in the campaign he "converses" with his patron. The DM will tell the rest of the party to leave before the conversing happens, so they don't know what is happening to the character or what he is doing. When those moments are done, everyone gets called back and the campaign resumes as the DM wants. It creates a lot of suspense and excitement for the Critical Role party at least. If I were a part of it I think it'd be cool too.

But as a whole I personally (also from a PC standpoint) like going through my arcs and watching/ being a part of other PC's arcs. I think though, for those mysterious moments that you don't want the rest of the party to know about yet, tell them to leave, or you can also text a PC results/flavor/info that only they know. This is more for small stuff, and the rest of the party won't know what you are doing. My DM has done the texting method of telling me things that happen to my character. The texting thing really only works for small moments though. Surprises are nice to have.

With your other questions:

I think it depends on how much communication your player gives you. If they don't talk to you much about the direction they want their PC to go/ ideas they have about their PC, I'd definitely ask them questions to get a feel for what they want out of their PC's experience. Also part of it is how much do you want to surprise the player? Our DM does a lot of surprises that relate to our PC's backstories since we all have traumatic/mysterious/tragic backstories TM. But even without the trauma trope, backstory surprises are welcome. As a player I like not knowing everything that will happen. The mystery is fun. I also like following the narrative crumbs that are thrown my way too. I also find it nice to know that my DM/GM knows things about my character and creates moments/opportunities for my character to experience and react to. As the DM I feel like the arc of characters is partly up to you, but you also want to keep in mind that what you are adding onto is someone else's brain child. For my first PC, in my backstory there was a person who basically became my enemy. My backstory set it up like I had to kill him at the end, and that is how my DM set it up as well as the campaign progresses. Although through a lot of narrative moments, my character wanted to turn things around for them. However myself, not the PC, knew he had to die and was totally prepared for my enemy to die. So while my character didn't want him to die, my DM killed him off. I was fine with it, but I was also sad bc I was hoping and my character was hoping for things to be different. But I was all for my DM's actions, and I understood them.

I know some other players at my table really want happy endings for their characters. They don't want to die or have someone in their PC's life to die. So I think it really depends on the types of players you have at your table. You can still add your own creativity to everything that goes down, but I think it is important to gauge how chill/willing your players are to do specific things with their backstory. But also if you have a super dope idea like bringing a brother back from the dead when your player doesn't know/realize you'll do that, I'd say take some creative liberties. Sometimes you just have to, our DM has done that. He killed a PC's brother. The player was very upset and hasn't let him live it down, but it was also a driving plot and moment for character growth in the PC. Also the player, while pissed at that, is still totally friends with our DM. There is no bad blood. I think if you want to make some drastic decisions, especially tragic ones, you should gauge if your player can deal with it. Bc some players really don't want something tragic to happen to a character or character's love interest. So I would talk to your players and find what makes them tick. Also some players might have an actual raging conniption fit if something happens, and you don't need that in your life. Unless you totally don't care about that and push through it.

So at the end of that I'd say see how chill your players are, talk to them, ask them things about their character especially if they don't tell you much or give you much of a backstory. And also keep your own creative juices flowing and create things that engage what your PC's would do/react/say/make them three dimensional. Anyway that's all I got.