IF you're saving directly to MP4, it does not. And that's no fault of OBS. That's the fault of MP4.
Save to FLV or MKV instead.
MKV and FLV both save as recording goes. You might lose up to about 5 seconds at the end of the file in the event of a crash, which is a lot better to deal with than losing the entire file.
Using either will not cause lag (In fact, lag ends up being reduced compared to mp4 in some cases)
Youtube can accept both last I checked.
No. USB drives have far lower speeds compared to internals.