So, divination. I haven't had to deal with it either, but I'm not terribly worried as long as my entire game isn't reliant on a single unknown element keeping the players from storming the final dungeon. I always try to have multiple villains so that the game doesn't fall apart if they discover one of them, and never assume any mystery will remain unsolved- indeed, if a mystery exists in the game, it hopefully WILL be solved by the players sooner or later.
Up until lv 5 spells and onward, most spells are limited in scope enough that you can get away with giving 1 piece of information per spell use, and that shouldn't unravel a mystery by itself unless they already had all the other clues, or are asking VERY good questions. After level 5 and Commune, it's a bit tougher, but that is the nature of high level casting in general. Still, by that point even if they know what to do, actually doing it (taking down the villain, going to an extraplanar location) can be tricky.
The biggest problem to me, then, isn't so much a player vs gm thing. It's the classic problem of spellcasters vs martials and skill users trivializing things, but that's a built in problem with the system and as long as spells exist that have a purpose to give players answers to questions, that's going to be a problem. It's only a major problem if the group as a whole feels it's a problem. I wouldn't want to sacrifice a cool thematic element by banning the spells outright though, unless it did turn into a reoccuring problem.
Well, I was attempting to account for the worst case scenario (since you were talking about divinations scaring you)- a player who does know how to ask really on point questions. I can imagine the potential information a good player could gain in the right situation being quite high. But even in that worst case scenario, not usually a big deal...
Unless the player is making it hard for the other players to contribute to investigations, which brings us to martial/caster disparity.
I don't want to make it an argument of the topic whether C/MD exists- but it's true that this is an example of something that only casters can do- ask the GM a question and expect a reasonable answer (within a certain margin of error defined by the spell). There's some nuance but it's definitely there. My only reason for mentioning it is that if you have a Rogue in the party who wants to investigate the old fashioned way, the diviner could step on his toes some. Which means it's not a player vs GM issue, it's a player vs player issue.
But, nothing inter-player diplomacy can't solve in a decent group.
Divination COULD be a game wrecker. At high levels there's this strategy called "scry & fry". Basically the party scries the BBEG and teleports in his lair avoiding 99% of his dungeon avoiding all the other encounters so they can face the final fight with all of their resources intact.
This can ruin your day as a GM UNLESS you know of the strategy and take measures to foil it. At worst let it work a couple of time, then warn the players it has become common knowledge the party usues divination and teleport to kill their enemies. Then have the baddies use the strategy against them while the party's pants are down, so to speak and see how the players like it.
Dream Scan is a problem if you know the information you need is in someone's head and they have to sleep. With enough time you'll eventually get everything you need.
The best way I've come up with to limit Dream Scan is have the target remember the questions when they wake up, so they have some forewarning and can change things up if they have time. You could also treat it like Sending where you recognize who is invading your dreams if you've previously met them.