Why does each Emu have a Cab?

I understand it is more flexible, but it seems like it’s more expensive and less space efficient. I suspect it has something to do with the FRA as PATH also has a cab in each car when it clearly does not need them. I believe the subway comes in sets of around 5 for the R211,
so when not regulated by the FRA there aren’t cabs in each car.

(Yes I know the New Haven line has unpowered cars but they are not super common.)

That is when an EMU is at an endstation. The driver can just switch off the cabin, walk back and turn on the cabin on the other side. Train is ready for depart after an mandatory brake test

It's not the FRA. Metra Electric has cabs and trailers. Same with South Shore, most likely same with CalTrans.

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The CalTrain EMUs are in 7 car Cab-5 trailers-Cab trainsets with full width cabs. This gives CalTrain the flexibility to run 7 or 14 car trains.

While the MTA M2s and powered M8s are married pairs, the MTA M4s and M6s were triplets with a cab-trailer-cab design, and unpowered M8 trailers without cabs. This gave the MTA the flexibility of running trains of any length greater than 2 car with the M2/M4/M6s and of running 2 car or any length greater than 4 cars trains with the M8s. This also means they can swap out 2 car sets for maintenance instead of taking an entire 5 or 7 car train set out of service for maintenance.

The new EMUs for NJT will have trailers with or without cabs, and the powered units will be cabless.

You could save a little money but lose flexibility by having fewer cabs or having cabs only on one end. Another reason to have fewer cabs would be if FRA safety standards were higher for cars with cabs (potentially front of train) vs cars without cabs (in-between cars or back of train).

Were M-4/M-6 "D" cars and the M-8 single units subject to locomotive inspections since motorized?

PATH's PA1/PA2 and PA5 "C" cars are known as motorized trailers, same with R44 and R46 B units (converted to
linked sets, except on SIRT).

The Lackawanna used a motor coupled to a trailer, both with cabs.

The M-8 single cars are not motorized although they do draw electric power for HVAC, etc.