How are you guys anchoring cabinets to the walls in a 7x16?

On a bare aluminum skin trailer the magnet trick actually should work, you just have to know what you're feeling for. The skin is attached to the studs with a line of screws or rivets every few inches, so drag the magnet slowly and horizontal across the wall and it'll tick on each fastener. Follow that line vertically and you've mapped the stud. If your results seemed random you might've been hitting the horizontal crossmembers too, or the trailer has rivets on the bottom half and screws on top, which throws you off. A stud finder set to deep scan can help confirm on the other side of the wall. For a bench that heavy I wouldn't trust skin screws at all. Two options that both work. One, open up the wall where the bench goes and add horizontal blocking between the studs, screw the bench into the studs and blocking. Two, the way I did mine, line the whole wall with 1/2 ply screwed to the studs, then mount anything anywhere with no hunting. Costs a bit of floor space but you never think about it again. Through-bolting with backing plates works too, just make sure the backing spans two studs or it'll work the holes over time. Gravel roads are genuinely hard on trailer interiors. I lost a couple of cabinet screws and a shelf bracket before I got serious about it. Blue threadlocker on the bolts is the move, it holds against vibration but still comes off with normal hand tools when you want to rearrange. Where you're using nuts, nylocs are even better since they don't care about oily surfaces. Skip anything red unless you never want that bolt out again. One more thing on the bench, put a lip or cleat across the front edge. Braking loads are what kill installs like that, and a 3/4 ply bench with a cleat tied into the floor as well as the wall barely loads the wall at all.

Another way to skin it if you ever redo the walls someday: make a cardboard template of the whole wall and mark every stud on it before the skin goes on, then trace it onto the ply and label it. Did that on my old job trailer and never hunted for a stud again. For now I'd say don't overthink the wall connection. That skin flexes more than you'd believe, so get the bench down to the floor with angle brackets and let the wall fasteners just keep it from tipping. Old timer I worked for used to mount everthing to the floor and swear the walls were just there to keep the rain out.

For what it's worth I went the other direction with my cabinets and made the floor do most of the work. Everything sits on a plywood base screwed to the crossmembers down low, and the wall just keeps things from tipping. cabinet doors were the real headache for me though, they'd swing open on every washboard stretch till I added a simple barrel bolt on each one. Better than any threadlocker I ever tried, and it's not like you're solving the vibration problem anyway, you're just containing it.

Gonna push back on the magnet advice a little. Half the fasteners holding these skins on are stainless or aluminum rivets depending on the manufacturer, and a magnet reads exactly none of that. Mine has stainless screws up top and I looked silly dragging a magnet around getting zero ticks. What actually worked was measuring off the frame - the verticals are usally on an even spacing from one end, so find one and count across. For the threadlocker question, blue is fine but honestly a plain bolt snugged properly plus checking it after the first few hauls has held everything in my box for years.

One thing about the gravel pounding that nobody's touched yet - tire pressure makes a bigger difference than people think. My 6x12 came with the tires pumped to max sidewall and every washboard section turned the whole inside into a maraca. I dropped them down to about 40 psi and the rattle inside cut way down. Doesn't fix a bad install, but it buys your cabinet doors and drawer slides some life.

Also on the backing plate route, I'd use fender washers on the inside face even with a plate. The skin is so thin it will oil-can around any bolt head over time, and a big washer spreads that out cheap.

Update: went back out with the magnet and dragged it slow like one of you said, and sure enough I got ticks along a couple lines. Found enough fasteners to trace two studs, so the bench is bolted through with backing inside now and it doesn't budge. Still stuck on my other question though. Do these things shake bolts loose on gravel, or am I overthinking the thread locker?