The house skip happened to me when I was randomly mashing start, but it's really inconsistent from what I can tell.

I always mash start once gameplay ends, so it might just be that and luck. I do remember I was changing around voice packs and stuff as well before that run, so I thought that might've had some effect.

If you need to restart a level is it OK to go to the portion you failed at or should we require that the whole verse is restarted? Ex: A solider falls out of map during a lock in fight and the first part of the verse was a camp sequence. Can I abandon and go back to the start of the part or do I need to restart the verse? Mainly thinking about NG+, but this can happen for NG as well.

Personally, I am leaning towards restarting the part. I don't see any merit to going through camp sequences again, and it can cost a ton of time.

As long as it's still counted to the final time then including the restart. Else it's not a standard run and is now segmented basically now IL format. If I'm understanding what your saying correctly. But in general it's always easier to restart a verse right away in most cases if it's a softlock or other.

Take this case:

Could I have skipped the camp scene since I already did it once?

yeah when you launched at them you stopped them from moving into range. It would have been faster to just do a restart on that verse. But you still take the time loss regardless since its not IL/Segmented.