screws
I'm going to be mounting a network cabinet on some studs. I was advised to use lag screws. (Not really sure what these are but I assume they are for heavy things being secured to studs).
Should I be putting washers behind the mount in front of the wall, or under the top of the head of the screw but still on the front of the mount itself (so it'd be Top Of Screw|Washer|Mounting Bracket Metal|Wall)?
The air gap problem is quite common: The sheetrock on many walls I have encountered doesn't contact the studs everywhere. So, when you tighten down a TV mount to the wall, the drywall deflects toward the studs. You can easily cause cracking of the drywall, particularly when there is a seam in the area.
The principle of the solutions in this case make use of a fastener's ability to support an overhung load.
The fastener and the building structure have to be strong enough to withstand the torque created by the downward force of TV and Mount acting over the distance between the drywall surface and the building structure.
Assuming you find wood building structure in there (furring doesn't count here because it will twist under the torque), I recommend hanger bolts.
Use a spade bit to clear a hole in the drywall big enough to accommodate a nut flush with the surface of the drywall. Then put the mount over the hanger bolt and tighten another nut down on that. The first nut will keep the drywall from deflecting toward the wall. Size the hanger bolt larger than normal because of the overhung load. From your description I would guess that 4x 1/4" hanger bolts would be adequate.
Make sure you pilot drill for your hanger bolt or you risk splitting whatever wooden structure you're screwing into.
I've also encountered furred out concrete clad in drywall (common in highrise condominiums). In that case, the same principal as Case 1.1 applies, but use "drop-in anchors" and a threaded stud instead of hanger bolts.
Sometimes drywall is attached to and is in contact with (i.e. no air gap) furring.
Where furring is attached directly to the building structure, drive multiple wood screws directly into the furring. If there is some hard material behind the furring, ensure that the screws are short enough. Otherwise they can jack the furring from the building structure when their points impact cinderblock or concrete.
The principle of this solution is to transfer the load to the building's structure directly through the furring.
Where furring and drywall are creating their own structure, drive multiple wood screws into the furring to fasten your TV mount.
The principle of this solution is to use the furring to distribute the load to a large area of the composite structure of furring and drywall.
Drywall itself is rather heavy, so if the wall is able to hold itself up, there's a good chance it can hold up your TV to provided you distribute the load over a large enough area of drywall and furring.
There is quite an 'overhang' of the screw (24 mm) from the wall. That's why you need at least 6 mm screws in diameter. It's head is about the maximum you have.
That implies 12 mm anchors, which are 60 mm long here.
The resulting screw would be 6x80, or 6x90, if you have longer anchors.
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Washers are almost always to 1) spread the load of the screw head to a larger area, and 2) to prevent damage to the face of the component being fastened. They can also be used as spacers, but I don't see a need for that here.
In your case, the washers go under the screw head and over the cabinet mount.