Can runner choose to take brain damage and prevent it on Black Level Clearance and continue the run without jack out?

My first thought was no, because runner didn't take damage if he prevent it.

I might be confused by Fairchild 3.0. "When runner can't pay, he must trash an installed card."

Thanks

Yes, because it's en either/or option. When presented with a binary choice like this, you make the decision, then deal with any resulting consequences of that decision, but the other option never gets picked (however, you MUST choose an option you are ABLE to fulfil). With Fairchild, the runner could also, for example, use a prevent effect to avoid trashing the installed card (such as LLDS energy regulator).

Compare this with a card like ****** and Grab, which specifically states you may take a tag to prevent a connection being trashed - the wording here means that taking the tag is a cost. If you prevent the cost, you haven't paid it, and so the thing happens.

So, the difference:

The first card, is choose A or B. Deal with A or B happening.

The second card is A happens, but you may pay cost B to prevent A.

Black Level Clearance falls into the first category.

I hope that helps, rather than confusing matters further.