Yes, in fact, you can gift the MQMs that you earn from the AmEx card. I recently learned that with choice benefits you can only gift one of each gift type, so gold status, for example, to one person not to two. So, my plan for this upcoming year since I fly enough BIS miles to earn DM with a lot leftover is to gift gold to my wife and then gift my AmEx MQMs to my oldest son who flies somewhat frequently but not enough to earn DL status on his own so that he will at least qualify as FO. This is a nice perk with the AmEx if you are trying to help someone else along their path to status.

No, you can not gift MQMs earned through actual flying.