Hi there!

On Runners's heap all cards are face Up ? When, as a Runner, I suffer net/meat damage or have more cards in the grip than allowed (due initial limit or brain damage) should I discard my cards open so the Corp knows what I had ? Or can I have face down card in the pile as well hiding my strategy or possible events that I had intention to surprise my opponent. ?

Thanks in advance !

Cards in the heap are discarded face-up. From page 6.

Heap
This is the Runner’s trash pile. The heap is kept adjacent to the
Runner’s identity card. This is where Runner cards are placed
when they are trashed or discarded. Cards in the heap are
faceup and inactive. Both the Runner and Corporation may
look through the heap at any time, but must maintain the order
of its cards.

The only part about that rule that makes me scratch my head is why the Runner's discards must be kept in order?

Because there may be cards (in the future) that let the runner get the top program/virus/event/whatever from his heap.

Ironburn said:

OP: There were cards that interacted with the heap (trash) in this way in the original game, and I suspect that will be the case again.