Answer: FLOOR MAT
This puzzle operates as a walkthrough of an escape-the-room puzzle, with the "spoilers" redacted.
Each redacted object is a word or a two-letter state abbreviation (with the exception of the "single bit" that falls out, which is the letter O). Each device (globe, oven, etc.) is a consistent operation applied to the letters of the redacted objects.
Taking these assumptions, you can trace the valid transforms that could be done to words, helped by the flavor and hinting in the puzzle narrative. The devices are:
Tracing the words through all the operations, you periodically put the results of your work into some slots. These slots are for FL, O (the "bit" that falls out early in the narrative), OR, and MAT, spelling the answer FLOORMAT.
Below is the full puzzle with the objects un-redacted:
You find yourself in a modern-day one-room schoolhouse. I guess this is detention? Whatever it is, the door is locked. There is, however, a large box by the door with four slots (this will come in handy later).
Looking around, you see a woodshop area, a home economics area, a geography nook, a music station, and a teacher's desk with a weird machine and a small safe behind it. There aren't any student desks, which seems a bit of an oversight...
First off, go to the geography nook and demonstrate the sort of behavior that probably got you detention in the first place. You knock the globe off the shelf (right onto North America!) and several small pieces fall off. The game lets you pick up two of them and now you have PA and AR in your inventory.
Not wanting to get caught, you try to fix the globe. You put AR back, but LA just falls right off instead. I guess you could say that this globe is pretty run-down now. Moving on...
Go over to the science lab area and pick up a jar of AMOEBAS. Take them over to the weird scrubber machine behind the teacher's desk. Put them in the slot with all of the sponges, and put the piece you just got into the slot with the glue. The object looks pretty weird after the first stage of the scrubber, but once everything gets stuck together, the LAMBS look fine (though having to keep little animals in this space is kind of annoying).
From the home ec area, pick up the LOOM from the textile table and take it to that weird device behind the teacher's desk. Put it in the spongey compartment of the scrubber and put PA in the sticky slot. Pull the lever and it all becomes a kind of tree. There's some kind of glitch in the scrubbing process and a single bit falls out. Catch the O and add it to your inventory.
Take the PALM out of the scrubber and back over to the home ec area. Put it in the left side of the two-chambered oven. Herd all the animals wandering around the room into the right side of the oven. Set the left knob to 1 and the right to 4 and wait a moment.
Open the left side of the oven and take out the BALM. So soothing and smooth! Now head over to the shop area and use the double-bandsaw on it to get AL. Yikes, is this really a good way to teach kids about safety? You seem to still have both of your thumbs, so take the piece you just made back to the globe. Maybe now you can fix it!
Try to fix the globe with it, and FL falls off. Well, you're definitely getting warmer—it matches the first slot in the box by the exit! Place it there now.
Now head back to the oven and take LAMPS from the right-hand compartment. Time for more dubious shop safety! Take them to the double-bandsaw and get a much dimmer but louder object.
You notice a SAW hanging out in the shop area, and its teeth are looking kind of rusty and grimy. So, take it to the scrubber and put it in the spongey compartment, and put the AMP fresh from the bandsaw into the glue slot. Huh, card games certainly weren't part of the curriculum when I was in school! Stick this SWAMP into the right side of the oven for now.
The safe is unlocked for some reason. Open it and take out some GATS. Yikes! The teacher must've confiscated those from some really unruly kids.
Put them on the turntable in the music area and get another animal (just one majestic STAG this time).
Flush that from the music area over into the left side of the two-chambered oven. Turn the left dial to 2 and the right dial to 2.
Open the left side of the oven and take out some SWAG. Take it to the double-bandsaw, getting WA. You know what to do with these things by now...
Try to fix the globe with it and get OR. Hey, this fits in the third slot by the door! Go put it there now.
Back to our cooking adventures. Open the right side of the oven and take the STAMP. Take it over to the double-bandsaw to get a TAM.
Head back to the music area and put it on the turntable. The jarring bagpipe music fades and you can take the MAT. This looks like it'll fit in the fourth slot of the box.
Now you can see that the little round piece in your inventory will fit perfectly in the second slot of the box. Put it there now!
The box opens. Ta-da! Take out the FLOORMAT and put it in the obvious place for it in the room. The school bell rings, class is dismissed, and you can successfully escape! Yay!