I disagree with this. I'm curious to see what happens in the faq. It's completely contradictory to the rules for downed players to be able to pass, tackle or sprint. It just doesn't make any logical sense regardless of how that skill is assigned to them. Down players lose all skills. The coaching ability assigns a skill to a player, but if the player is downed they lose that skill.
No, downed players lose the remaining skills/abilities on THEIR card. Coaches add to the player even if they are down.