So, while my application is formally processed, should I try to informally reach out (e.g., ask more details about the role via LinkedIn) to the hiring manager and/or future boss — this, in order to have my application noticed?
No.
Companies (and most hiring managers) frown upon folks trying to go around their formal processes.
And no hiring manager I know wants to answer questions from a stranger that hasn't even passed the initial screening phase.
Just be patient and follow the company's process.
P.S.: With hindsight, what I want to know is whether this move would be perceived positively as:
"Wow! Here is a candidate showing they're really motivated for that job and goes beyond simply filling yet another form/who's not blindly applying to any offer coming! (So let's see what's they're really like)"
Or whether it would be frown upon like:
"Who does they think they are?! Here is a candidate that's trying to jump the queue and doesn't formal processes matter! (So let's not bother considering their application)."
They may just ignore your attempt to "reach out". But of the two choices, the latter is far, far more likely in my experience.
It's not a risk I would suggest taking.