Ok, so you've shown that the devices attach and sync to other computers so the devices are probably OK. I have a 2013 Macbook Air I7/8GB/500GB and an iPad 3 and an iPhone 4s and my devices attach and sync properly, so it's not the case that they're incompatible or anything -- that would be pretty silly anyway. It could presumably be the cable (you probably used the same one every time?) or the USB port or the CPU board or the software.
If you swap out the cable you can eliminate that. You could try the other USB port to see if that makes any difference and if possible remove any other USB device. If it still doesn't work, it's probably hardware. If you bring it to Apple they'll probably swap it out depending on whether you have a build to order device or not. Assuming that is your intention, you'd want to wipe off your data anyway, so make a good backup (or better 2) and then try an archive and install (using the recovery partition --shut down, power button and immediately even before you hear the tone CMD-R. Then try installing OSX over your existing OSX). That replaces all the OSX files (or so they say). If that still doesn't work since you would want to remove your files before trading it out at Apple, do an erase install. Same as previous but use disk utilities in the recovery partition to erase the disk and then do an install of OSX. Now you're back to factory original settings. If it still doesn't work, try to get Apple to replace it.
If it works with a different cable, you have a bad USB cable. If it works in the other port, you still need to replace it, but you probably have a bad port. To make sure, if you unplugged any other USB devices, plug them back in and see if that makes it not work again. if so, something about that device combination is a problem. I think that is unlikely.
If it works once OSX is reinstalled it probably means you've made some change to your system that made the sync stop working. You can try reinstalling your stuff one at a time till you discover what it was.
If it works once you do a erase install, well then maybe something in your user settings might be the problem? Again you can try installing stuff to try to determine what happened. Or you can migrate your settings and see if things stop working.
Keep in mind that erasing the disk erases all your files, so don't do it unless you have at least 1 and preferably 2 good backups. If you take it to the Apple store and they swap out your computer they will NOT transfer your files!