According to the manual, the PSR-E313 has MIDI ports rather than USB ports, so you do need a MIDI cable, as well as a MIDI-to-USB adapter. I haven't had to use one myself, so I don't know what's involved, but I expect you'd need to install a driver to go with the adapter. If you've been able to play MIDI files on your computer, and use the computer's keyboard to send MIDI note messages to your PSR-E313, then I'd say the connection is probably okay and the problem is elsewhere.
When you play your Yamaha and send it to your computer, there's no audio involved if you're using a MIDI connection, just MIDI data, so you won't be able to hear anything on your computer unless you're sending the MIDI data to a DAW (such as Mixcraft) and have a virtual instrument assigned to the track. In that case you can record audio, but the sound is being produced by the DAW's virtual instrument rather than by the Yamaha keyboard. If you record the MIDI data from the Yamaha then you can edit the MIDI data in the DAW-- e.g., correcting places where you hit the wrong note by mistake, quantizing the notes so they line up exactly on the beats, half-beats, quarter-beats, etc., copying and pasting measures that you want repeated, and so on-- and once the MIDI track has been saved you can even change it later on to use a different virtual instrument. So it can be extremely useful to record the Yamaha's MIDI data rather than its audio output.
On the other hand, you can also record the Yamaha's audio output if you'd prefer, so that you're getting the Yamaha's "voices" rather than the sounds generated by an external virtual instrument. To do this, you must connect your Yamaha to your computer using an audio cable rather than a MIDI-to-USB cable. One end of the audio cable gets plugged into the Yamaha's headphones (audio out) jack, and the other end of the audio cable gets plugged into the computer's microphone (audio in) jack. You'll probably also need an adapter for the end that goes to the Yamaha, because the Yamaha's headphones jack uses a bigger (1/4-inch) plug, whereas the computer uses a smaller (.35mm) plug. You might also want to get a splitter for the end that goes into the Yamaha, so that you can plug a pair of headphones into one of the splitter's jacks, and plug the audio cable into the other jack. Then the plug end of the splitter goes into the .35mm-to-1/4-inch adapter, which in turns goes into the Yamaha's headphones jack.
As for playing MIDI tracks in Mixcraft and sending them to your Yamaha, you'll need to insert some Bank Select and Program Change messages at the beginning of each track to select the voices that you want the Yamaha to use. The list of voices in the PSR-E313 owner's manual has columns that show the Bank Select and Program Change values you'll need to use to select each voice-- except there's a slight wrinkle you need to be aware of. MIDI messages uses Program Change values that range from 0 to 127, but in some software the Program Change "program numbers" or "patch numbers" may be numbered from 1 to 128 (apparently based on the theory that human beings are too stupid to be able to count from 0 to whatever the way that computers do). In fact, the online MIDI documentation from the MMA (MIDI Manufacturer's Association) even lists the 128 General MIDI instruments using the 1-to128 numbering system, despite the fact that the actual MIDI messages use the 0-to-127 numbering! Anyway, what this means is that when you look in the voice list to get the Bank Select MSB and LSB values, as well as the Program Change values, for the voices you want to use, you must subtract 1 from the Program Change value that's listed in the manual.
For example, suppose you want to use the PSR-E313's voice 47 ("String Ensemble") for one of the MIDI tracks. The voice list in the manual shows that this voice is selected by using the values 0 (in the Bank Select MSB column), 112 (in the Bank Select LSB column), and 49 (in the Program Change column). When you select these values in Mixcraft, you must actually use 48 for the Program Change (i.e., just subtract 1 from the Program Change value shown in the manual), otherwise you'll end up selecting a different voice than you'd intended.
As for the other things you mentioned, the PC Mode function is used to quickly and easily change the settings of five other functions (Local, External Clock, Song Out, Style Out, and Keyboard Out) without having to change them individually, so you don't need to use the PC Mode function at all if you'd prefer to set those five functions separately (which gives you greater individual control over them).
The Local function determines whether the keyboard will produce sound when you play it. If you want to use your PSR-E313 as a MIDI keyboard controller with a DAW and an external sound source (virtual instrument) then you'll usually want to turn Local off. But if you want to record the audio from your Yamaha then you'll want Local to be on.
External Clock controls whether the Yamaha uses its own (internal) clock for the tempo and the beats of each measure, or whether it uses the clock signals from a DAW or other external MIDI source.
Song Out controls whether any songs you play back on your Yamaha will send MIDI data to a DAW.
Style Out controls whether any auto accompaniment (style) you play on your Yamaha will send MIDI data to a DAW.
Keyboard Out controls whether the Main/Dual/Split voice channels will send MIDI data to a DAW as you play.
I'll see if I can write up some basic tutorials on how to use the PSR-E313 and other Yamaha keyboards with Mixcraft to (1) record audio tracks, (2) record MIDI tracks, and (3) send MIDI data from Mixcraft to the Yamaha. I have Mixcraft Pro Studio 6, which hopefully isn't too different from whatever version you have.
Edit: I forgot to mention the Init Send function. This is used to send the keyboard's settings to a DAW so that the correct voice settings get recorded. For better or for worse, Mixcraft strips out any SysEx events from the incoming MIDI data, so if you do use Init Send some of the settings will be ignored by Mixcraft.
