I've owned or used all the machines suggested in this thread. I think the FE2 is a better machine than the FA. The meter readout on the FE is almost usable (no LCD to get nasty over time, but impossible to use in the dark) and the top plate on the FA is plastic (as opposed to metal on the FE2). Other than these and matrix metering, however, they are very much the same. The FA also takes the same motor drive, but it also takes the rather nice MD-15, which uses the AA's instead of the battery in the camera. I'd go with the FE2 if these were the only choices. The viewfinders are marginal at best if you have glasses, though.
The F3 is quite a bit bigger, has a slower shutter and sync speed, and has the same kind of readout as the FA (I.e.: not that hot, but it at least has a light, albeit with a very crappy button and a flaky circuit). The wind stroke may be smooth, but it is also obscenely long (i.e.: like 180 degrees to the 135 degrees of the FM/FE or F2... shoot them side-by-side and you'll notice the difference, and it's a pain IMHO). These cameras really shine, however, with the MD-4. KEH has F3HPs (fantastic for glasses, and the best of all MF Nikons) for $99 for Bargain grade (bit scuffed, but fully working, less if there's an engraving) with a 6-month warranty. Hard to go wrong there, actually. Oh, as noted, they made a million of them. MD-4's are in the $10-20 range. A bit heavy, but absolutely rock-solid inside and out. There's a reason they outlasted the F4 and almost through the F5.
The FM2n has the best display (an LED won't die like an LCD, and you can actually see it in the dark). Sure, it has no auto, but you technically don't need batteries for it EVER. Just get used to seeing your light and you won't even need to meter. Use Sunny 16, subtract 2 or 3 for open shadows or a cloudy day and learn what the lighting is in your favorite bar (1/60 @ f/2 w/ ISO 3200 for mine, plus 2 stops for behind the bar, minus 1EV at the dark end and 1/125 @f/2.8 in the booths, for example).
As far as ISO 400 film at f/16 or f/11, unless you're shooting on a beach or sunny field of snow, you will never need more than 1/1000th of a second. An FE would be fine (I like mine quite a bit) and also takes older non-AI lenses without modifications to either camera or lens. Not as many repair parts as there used to be, but they're still fixable (ask Hexar RF owners about that on a camera 20 years newer by a smaller manufacturer). The FM has the same advantages as the FM2n, just a slower shutter.
The FM3a will cost you 4x as much as an FE2 or FM2n. While it's nice, it doesn't add anything but all-mechanical speeds to an FE2 (which isn't exactly a battery hog, BTW). It's well outside your $160 budget, so don't worry about it.
The FG, while it doesn't have a DoF preview button, is a swell little machine. Very, very tiny and pretty quiet, too. The meter display has LEDs, so you can see it in the dark. Pair it with a Series E 50mm and you have a cheap, quality machine which you won't mind if it gets a little dirty or dinged.
I liked the F and F2, myself, but the cameras are a bit bigger and the metered heads are huge (and often not working). Plain prisms (especially for the F2) are obscenely expensive (as in $400 or more if it's a nice one or black), but are the best users now.
DO NOT think of buying a camera to "fix it up" as a way to save money unless a) you're experienced in repairs, and b) you have another camera to shoot. Junk cameras are often just that -- junk. Never expect an eBay seller to be fully honest about all the problems. If they say there's one problem that renders it inoperative, assume it's scrap from a number of problems and pay accordingly.
Well, hope that was helpful.
--Bob