The Withdraw action is a special action but doesn't actually require that you move a full double move - it merely means that you withdraw from the combat (and mechanically that the first 5' you leave you leave in a way that does not provoke attacks of opportunity. You can then move as much or as little as you want after that first 5' of movement and you do provoke from that further movement. So as I noted there actually isn't any "extra" movement in a Withdraw action
As a GM if your mount moved a double move I would indeed require you to take the ranged penalty (modified by your feats etc).
In terms of whether a mount could take this action as a GM I would rule as follows:
Does the mount know the "Down" trick? (i.e. the trick to break away from combat)?
If the mount is NOT an animal companion - then it is a MOVE action to handle animal to do a trick that it knows (DC is 10 or 12 if the animal is wounded) or it is a FULL ROUND action to "Push" the animal to a trick it doesn't know (DC is 25 or 27 if the animal is wounded). Taking a full round action would clearly prevent you from also attacking, taking a move action probably wouldn't prevent you from attacking (assuming you already have your weapon out and ready).
If the mount IS an animal companion (or mount for classes that get them) then it is a FREE action to handle animal for tricks the animal/mount knows and a MOVE action to push the animal. And you get a +4 circumstance bonus on your Handle Animal checks for your animal companion.
Mounted combat can be complicated with a lot of feats that may also impact how combat resolves. I'm assuming the character in question has mounted combat feats. If the character doesn't have the Mounted Skirmisher feat then they are restricted to a single attack action if the mount moves more than 5' (if they do have that feat then they can take a full attack action even as their mount moves) and Mounted Archery would impact the penalty they take if the mount does move a double move (or the higher penalty if the mount runs)
Ride skill checks depend a bit on whether the mount is combat trained. Assuming it is combat trained you technically need to make two checks to do mounted archery - a DC 5 check to 'guide with your knees" and a DC 10 check if your mount is fighting and you want to fight as well (for most mounted characters the DC 5 is nearly automatic so I rarely see it). A non-combat trained mount takes a move action to control in battle (DC 20)