Essentially, my question is that are we allowed to play through the game to get certain items to then do a single segment? Or do we have to start from the main menu and use whatever they give us?

And, for the latter, would this mean that if someone did a full campaign run, then that means their segment couldn't count even if it were the fastest, right? (excluding the first level obviously)

This is an interesting question.

I personally look to individual level runs as a preparation for full campaign. So I do them the way I plan to do in campaign run later. Of course get 'prepared' heroes will give you an advantage. But we can not avoid it, because we can not set up the 'right' items sets. So it should be either none or any. None is a sick option because you will need to "clear" your heroes after every mission in IL runs. So, no items or preparations ban looks like the better option.

Of course you will never be able to do full campaign with the time which is the sum of your ILs runs if you use any specific preparation

Ok! Thank you for the response. I think it would be better to just leave it as "whatever goes" kind of thing.