Heh, this last post made me smile because in the PbP AW game I' running, there's an armored boat which is the scariest thing in town.  Of course, the Show just got the crew to panic with a really good world-breaking song and the boat ran aground in a skyscraper... and that's just the beginning of the problems this damn boat will create! I would also note that even if the PCs are able to grab hold of an actual no-fooling TANK, a tank is not a rental car; unless it's some kind of high tech robot tank, it requires a gunner, a driver, a loader, and a serious mechanic to keep it rolling.  Sure, I'd let a PC sneak into one and maybe fire off a shot, but there's a lot more to it than just jumping it and shooting up everybody else in the world.  Enforcing that kind of realism, that kind of friction on what the players want to do, is an important part of making the world seem real.

But seriously, don't try to keep the players from doing the stuff their playbook makes them really good at.  That just makes you frustrated and the players annoyed. The Gunlugger character is inevitably going to solve problems by shooting them whenever possible, from a great distance if possible.  That's OK!  They're going to do that, just like the Hardholder is going to order his men to seize control of stuff and the Brainer is going to figure out creepy stuff to do with his violation glove.  If you follow the game's agenda for the MC and keep making soft and hard moves, they'll find plenty of places where their sniper rifle isn't helpful. 

This can be problematic when the player isn't really addressing the situation other than through violence, in other words playing AW like a roleplaying version of Grand Theft Auto, but I think that's more of an OOC issue. 

My favorite trick for an NPC Hardholder: "Hey, PC A, the cardholder just asked you for a way to neutralize the threat posed by PC B if they ever go off the reservation.  Just an insurance policy. What do you do?" Get the Savvyhead working on a gadget to disable the Brainer's gadgets, or the Brainer trying to put a kill-switch in the Gunlugger's head.  It doesn't have to be mean, it doesn't have to be a secret.  It's just, "I gotta protect the settlement, and you're a major threat." 

I wouldn't interfere with the Skinner player's use and abuse of their moves. After all, a Brainer can explicitly keep puppeting someone, and the Hocus can wield unnatural influence over a whole gang.
Plus there's all kinds of things you can do with the setting and how other NPCs feel about their buddy Deek being under the Skinner's spell, neglecting other responsibilities, doing bad things to fulfill the Skinner's whims, etc.

Also, the character can always miss a roll even with a +3 Hot, and then Deek gets 2 hold on the Skinner. That should be fun!

In general, I like to let the PC's have their sick OP moves as written. If they do abuse them, that makes life easy for the MC.

I see this as totally legit, it comes with the territory of being a prince. 

This is the scene that appears in like every Hollywood movie about cops or soldiers where the lieutenant confronts the old captain and says you're doing this all wrong, you're going to let the bad guys get away or get us all killed. Then the older guy's jaw tightens and he tells the lieutenant to stand down and follow orders, and the younger guy salutes stiffly and walks out looking pissed.  It's pulling rank, basically. It might be a good time to make some notes about the pissed off young lieutenant who's going to start talking to his friends about this bullshit. Maybe even start a countdown clock if the Hardholder's been doing this a lot. So, like with everything the Hardholder does, you solve the obvious problem in front of you but you might create a much bigger problem behind your back. 

Yeah, players should read a sitch only when there's something going on. That's a low bar to cross in Apocalypse World, of course.  You can always do the thing they do in old Westerns and say, "It's quiet... too quiet."  You can totally invent stuff but I usually keep it relatively mundane, i.e. things they could plausibly observe. 

If the player wants crazy mystical tactical visions, they should take the Gunlugger Move "Battlefield Instinct" which lets you roll +hard to open your brain during a battle.  I love that Move.

In some Apocalypse World games, scavenging is a possible way to make a living. I ran a game which took place in a small community outside a very large dead city... there was a gold rush sort of atmosphere with prospectors going out, coming back rich or crazy or dying or getting bushwhacked.
Ironically, the PCs were never ever scavengers, they were trying to get by in a post apocalyptic boomtown.

If I were to do it again, I would increase the weirdness and danger associated with scavenging... make it more like Stalker than like D&D.

Yes and no.  There aren't "dungeons" in AW, just different places that are all dangerous and full of dangerous people.  You can certainly do a road-based game where the characters travel to different locations to take care of business and maybe even explore dangerous ruins in the blasted desert... but you're supposed to make things seem real.  Even if you're using some adventure-game elements, be sure to run them through the MC's agenda. 

Actually, a little bit of adventuring can be a great way to kick off new plot elements; the Gunlugger and the Chopper can gun down a lot of people and take their stuff, but the MC gets to determine what happens next and maybe they won't like it.

I think the main downside of this "adventuring" mode is that it takes the characters away from the web of relationships they establish with NPC characters.