Fall behind in or into?

Into. Through is also possible.

Could you please explain it?

"into" in this sense means "at the moment of time from the beginning of something". For instance, you could say

He smashed his car 30 seconds into the video. (not 30 seconds in the video)

So, it means "He smashed his car at thirtieth second of the video" and "The runner fell behind after reaching half way of the race and was never able to catch up with the other runners." can be replace with the original sentences. Right?

Yes, at the thirtieth second.
"halfway" is an adverb so you can't say "reaching half way " . You would have to say something like "after reaching the second half of the race, but the original sounds more natural to me.

Thanks for "the"
And yes the first form sounds better.

Thanks again.