Playing short form requires several shifts in technique and approach. My group typically does 2-3 hour sessions. A 4 hour session is a marathon for us.
Drop the Filler
The first thing to do is let go of filler material. Filler material includes setting up adventures that are "clue to clue to clue to oh actually interesting development". This is the default for a lot of play, and it wastes time. Let go of having the players play out haggling for everything, or explain each store they go to for supplies, and so on.
If there are encounters that would take up time but not actually provide interesting choices or hard implications? Cut them out.
There's also a lot of time wasting in putting in situations that make it unclear what the players should even be trying to do. "Where should we go next?" "I guess we keep searching until we find a clue of what to do next?" etc. Videogames used to use mazes to add extra hours of gameplay as a filler device, and this is the same kind of thing.
Let players get to the fun interesting stuff without this obfuscation and play not only goes quicker, it's more fun period.
Scene Framing
Scene framing is the next step. You no longer say, "Ok, where do you want to go next?" as the default question. If you know the general thing the players are doing, you skip up to the next part and get right into it.
For example, you know the players are trying to track down a cult:
"After three days of talking with shady people, listening to wacked
out babbling of street preachers, and having to do some small trading
in hallucinagens, you finally find out where they're having meetings.
At dusk, you find yourselves outside an abandoned church where several people died of plague years
ago..."
Think of how movies or tv shows will give you a montage - so you can skip the legwork and go RIGHT to the interesting stuff. This alone cuts out a LOT of time wasting material.
The Rules you use matter
If you're playing a game where combats take 45 minutes... well, one combat will take up a significant chunk of your time. Understand that many of the older games and traditional games expect you to be playing for 6-8 hours as session, so if they expect 5 combats in a session, and you've got 3 hours to play... it's not going to fit.
If you're playing a game where conflicts of all types, take 5 minutes? Well, then things move much quicker. So pay attention to what rules you're using and what their impact is, so you can plan appropriately.
Getting into character, exploring the world, etc.
I've been playing this way for several years now. My group gets into characters quite well, because the focus of our play is on characters, their issues and personalities, which is because a lot of our game time is putting the characters into crisis points and fun interactions.
I find a lot of people used to playing with lots of time, think the solution to character or world development is simply pouring on more time, when in fact, it's about directed and focused play. I've had several people say, "We've done more in this two hour one shot than I've done in entire campaigns, for years."
That's because I cut out the filler, focus on the characters' choices and actions, and try to give situations that are at turning points. Each scene should matter and have some impact.
You're asking for more specific rules about how Tongue of the Sun and Moon works, but you've already quoted all of the relevant text. There are no more relevant rules.
This is pretty much entirely in the realm of "Ask your DM". 5th Edition is a lot less precisely specced than 4th, or even 3rd, and this kind of question about what happens in situations where something isn't called out directly in the rules is supposed to be handled by the DM. There are no mechanics for how "touching the ki of other minds" works on a spiritual level, so this is well within the bounds of DM rulings.
That said, in one of my games, I'd allow you to do basically whatever you wanted to do with that ability. I feel that class features should always be benefits, not hindrances, and that creative play should be rewarded. Want to talk to some people, but not others? Great! That sounds like an interesting and non-linear way of using that ability. Want to purposely not understand someone speaking Demonic, since they may be cursing you? That'd work too.
For your third question: I'd run it that you don't have to know that someone is specifically listening, if you don't want to. Like, you could say something to an apparently empty stadium filled with invisible people with the intent that everyone who can hear you can understand you, and everyone would understand you, even if you don't really know that they're there.
For your fourth question: You're not really affecting people with anything, per se. Since there's no mention of a range, or any kind of targeting constraints, I'd run it that you could speak and understand anyone, anywhere. If someone broadcast a message on a TV in a different language, I'd let you understand it.
Best Answer
Characters can tell time about as well as any normal person...
(and about as well as it's interesting for them to know)
Barring a a session of video games or something else very interesting, you and I know about what time it is.
Sometimes you lose track of time, but it doesn't come up a lot. Granted, I don't know off hand what the exact time the sun rises or sets, but I can make do.
Then, what is the benefit of this ability?
DM judgment. The only real time this would come up is if the DM wants it to come up. The DM can create some sort of complication or constraint where proper time keeping is important.
Examples include (but are not limited to)
This is also an opportunity for the DM to empower players. If one of the players at my table takes Keen Mind, I'm going to be looking for ways to make exact time telling important, or knowing where North is.