[RPG] move 240′ and attack in one round by using two riding horses

Given that you appear to have all the rules already, it seems that the problem is in interpretation. Given that, my view is :

1. Should the mounted PC be treated as a normal mount for a PC, losing his turn and becoming subordinate to the rider?

   • Answer: No. The mount is the same level and equal in power

2. If 1 is "no", do they share initiative counts?

   • Answer: No. They each roll initiative, and act on their turns. If one wants to move and the other doesn't, then the rider either dismounts (if the rider is the source of the move) or falls prone (chosing not to move with the mount). As a generous DM I'd allow an acrobatics check to avoid falling prone.

3. If 1 is "no", can the rider use his move action to make the mounted PC move, or force the mount to act by spending his own actions in any other way described by the Mounted Combat rules?

   • As above; if one wants to move an the other doesn't, then it's either a dismount or a fall-prone for the rider, depending on who's turn it is.

   • Per the comments; the mount should only be able to be move normally once per round - whichever commands the mount uses their move action to move, the other still has a move action but can't use it to move (unless it's the rider dismounting). Clearly other sources of movement (e.g. warlords) would still have the expected result - and the mounted combat rules cover that sort of thing already.

4. Can the mounted PC use the rider's base skill check bonuses as described in the Mounted Combat feat (e.g. a human with a pixie on his shoulder uses the pixie's Stealth modifier)?

   • If the rider has mounted combat and the mount is willing, then I would go along with this.

5. How would feats like Holy Steed, which gives a rider's mount bonuses to defenses and damage, be applied?

   • I would go along with this. As a DM, I'd much rather my players bunch up than spread out. A small bonus is worth the penalty.

The mount and the rider each have their own full turn with movement and action. When a player mounts an animal and decides to control it:

The initiative of a controlled mount changes to match yours when you mount it.

The mount's position in the turn order changes but it keeps its own turn on which:

It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

Therefore the mount can move its full speed and finish its turn. After that the character takes his own turn on which it can dismount using half of the speed and move for the remaining half. What the character cannot do though is to mount, ride, and dismount as it would require two character's turns.

There is one thing that requires clarification though. If I understand correctly, ties in initiative are resolved once and for all, which means that after mounting the player has to decide whether it is the mount or the character who acts first during the round.

The only thing I see that may hurt my reasoning is this sentence (emphasis is mine)

A controlled mount can move and act even on the turn that you mount it.

It makes things arguable, but for me everything else is solid enough to consider that there is a mistake and it should be “round” instead of “turn” in this sentence.