I cannot say much about the NEX-5R, but these are general flash topics on which I can speak. And I have looked at the online NEX-5R manual to verify.
Generally many cameras have Auto modes, and they have A,S,P,M modes. Auto is fully auto, and A,S,P,M are not (not fully auto). For example, P mode is automatic exposure, but it will NOT be auto ISO and Auto White Balance (unless you turn those on yourself). It will not be Auto flash, but you can turn the flash on yourself. Auto mode will be Auto everything, no matter what you do. (OK, there is a Flash Off setting, like if at the Grand Canyon or in the football stadium).
See manual page 70. AutoFlash is Auto, it works automatically in Auto modes. It is not a factor for A,S,P,M modes. Auto mode pops open the flash automatically when it deems necessary (meaning of it fires in dark environments), and Auto makes all the decisions. Flash compensation should still work. A,S,P,M modes do not open the flash, that is, you open the flash yourself when and if you choose to use it.
TTL flash is automatic flash exposure in any camera mode, including Manual camera mode. Flash compensation is how you control what it does.
Fill flash is balanced mode, to add a little flash fill to the pictures, to make shadows less harsh, but specifically is NOT a full metered flash, which that full flash addition would overexpose an already properly metered ambient exposure of near subject (meaning of balanced). Fires all the time means it will if you select it. Fill flash implies in a brightly lighted ambient, something to fill. Fill does not imply a dark scene that needs a lot of flash to illuminate it. Not saying it will not do that, it surely meters to see what is necessary, and probably tries, but the idea of Fill is fill of a bright ambient.
Slow sync - when in dim ambient where flash is needed, there is no point of using the slow shutter speed that the dim ambient probably metered with the flash off, like say 1/4 second. That is why you are using flash instead. So if using flash, cameras in A and P modes honor a Minimum Shutter Speed With Flash (surely Auto too), probably 1/60 second Minimum shutter speed (you can set about any shutter speed in S or M modes). This 1/60 second is not about metering or flash or ambient or about adjusting anything, it is just a simple Minimum, always the same, regardless (not much intelligence in it). It will more seriously underexpose the background distance that the slower shutter might not. It can be a faster shutter if the ambient meters higher, but it won't be slower in dim scenes where flash is needed. It is a Minimum Shutter speed with flash. In the said dim ambient, you can surely see shutter speed jump up to about 1/60 second as you open (turn on) the flash. But it won't change if Slow sync.
So what Slow Sync is - is that if preferred, you can have the camera use the slow ambient metered exposure anyway, instead of jumping up to a Minimum shutter speed with flash (about 1/60 second). Slow ignores the minimum. This Slow shutter will then more normally and brightly illuminate the ambient, regardless of the flash. "Clear image of both subject and background" means the flash covers the near subject, and the slow shutter speed takes care of the dim ambient that the flash range cannot reach.
Slow sync is NOT a choice in Auto mode (page 70)... Auto is Auto.
Auto is only a choice in Auto mode.
Flash Off is not a choice in A,S,P,M, you would instead simply not open the flash door.
Rear sync flashes at the end of the shutter duration, to locate the slow ambient blur to be trailing the subjects motion (looks natural), instead of leading it (unnatural). Like maybe a moving car at night, Rear makes its tail lights follow the car instead of lead it. You will also need Slow Sync to cause the blur that you hope to position. If no blur, no point of Rear Curtain. Slow might be automatic with Rear? It is on Nikons.