You just have to take a look at any modern navy to get an idea. No ship runs smoothly and perfectly every moment of every shift. This is why the Enterprise-D had over 1,000 people on board. There are always crew monitoring systems and subsystems, running program simulations, maintaining the flow of energy, monitoring communications, running mission profiles, doing other research relevant to a mission, etc. Of course, there is down time and bridge banter occurs, like in any office or what have you.
Federation culture is no different than our modern culture in most respects. People do much of the same things we do here despite the higher level of technical sophistication. It's just that materialism isn't as big a thing as it used to be. Culture is still a thing though, and an important one by most indications, so artistic installations probably still exist. It's just that you hardly see them because the primary focus of the various shows is on the elite military grads who've won a high place in the workings of Starfleet and the Federation. Very rarely do you see what average folks on the ground are doing with their time. It's just like today. Politicians make the news and joe schmo begging for change with a flute or a bucket drum on the street corner doesn't.
In fact, Star Trek is kind of a dodge when it comes to predicting the future because there are no guarantees that the future will be as bright as Roddenberry envisioned it. Federation culture mirrors modern Western Culture so closely because that was considered the ideal culture to depict on television in the 60s, and by ideal I mean to say what would make the most money given the uncertainty of the premise at the time. Most people aren't attracted to things they can't understand, which is why so many shows fail before one full season airs. And Trek was by no means a guarantee at the time. Most fans today don't understand that Trek didn't actually do well in the ratings at first and faced cancellation more than once. It didn't start gaining serious steam as a property until it went into reruns in the early 70s.
As far as anything staying secret, part of Federation operations is to ensure secrecy against telepathy. So much like secret agents today are conditioned to resist torture if captured, anyone working for the Federation who is privy to vital top secret material is psychologically/psychically trained and conditioned to resist mental intrusion. And that doesn't even begin to cover how they are constantly monitored to ensure they aren't compromised. Cracks occur, of course, and problems are corrected. For instance, in sub-canon it was explained that Geordie was forced to accept ocular implants because he had been compromised via his VISOR one too many times for the comfort of top brass. They said 'get the eye implants or accept a less sensitive posting in Starfleet.'
There are plenty of things I've wondered about as well, having been a fan of the show since the late 80s, but as I've grown I've come to understand that explaining things in-universe to such an involved detail is kind of pointless. Sci-fi has always been a medium for exploring the human condition and stimulating discussion on how humans live and behave in the present. The best Trek stories adhere to this model and are in no way about how strong phasers can fire or about how fast a warp drive can run. The technical stuff isn't really that interesting to begin with unless you're an engineer/scientist trying to make stuff like that work in the here and now.
