If and effect discards more cards from opponents deck how are they discarded? All at once ore one by one? I think this matters most with Euron's Mongrel. I dont know if my opponents first shows all cards discarded by one effect e.g. Raiding fleet - 8 cards and then I can choose which one will trigger Mongrels effect or if I have to triger it right after the card has been put to discard pile.

Also it seems logic that Motley crewman discards only one additional card for each efect eg. Raiding fleets 8+1 not 8+8, right? They by this logic from my example above it would be the first choice.

When a single effect discards multiple cards, the effect only resolves once. As such, all cards are considered to be discarded at the same time. However, each individual card discarded creates an individual Response opportunity. This is really no different from the fact that when multiple characters are killed by the same effect (like 2+ military claim or Valar), they are all considered to be killed at the same time (when the effect resolves), but each creates an individual Response opportunity (for both standard Responses and saves).

So in your example, all 8 cards from Raiding Fleet are technically discarded at the same time when its Response resolves. That resolution creates 8 individual "after a card is discarded from a player's deck" Response opportunities. So after the full resolution of the effect, you have 8 separate Response opportunities to choose from for Euron's Mongrel. You can choose any of them, regardless of the "order" they landed in the discard pile, when triggering the Mongrel's Limited Response.

However, it ultimately makes no difference whether you look at this as "8 cards discarded at once, creating 8 individual Response opportunities" (the way it technically happens) and "8 cards discarded individually from a single effect" (the way that many people will intuitively see it) because of the timing structure. Because the 8 cards are being discarded by a single effect, you cannot interrupt the effect between cards - the same way you cannot interrupt a 2+ claim intrigue challenge between discarding card #1 and card #2. So if you want to Respond to any of these 8 cards with the Mongrel, you sill must wait until the entire effect resolves (ie, card #8 is discarded) before you can legally trigger any other effect. The 8 individual Response opportunities are all open and available at the same time.

As for the Motley Crewman, he specifically adds 1 to the effect. So again, whether you look at this as "8 cards discarded at once, creating 8 individual Response opportunities" or "8 cards discarded individually from a single effect" makes no difference. There is only 1 effect, so there is only 1 additional card. No matter how you look at it, the total would be 9, never 16.

I saw the title of the post and the smart arse in me said "by hand" But ktom's response is much more informative. Though I am a bit confused why he didn't explain the use of the supinator muscle in discarding