If Anima allows you to move/attack/move, then you can consider yourself the equivalent of a sniper with melee attacks. The enemy would have to figure out tactics that can trap/hurt you when you get in range. These might be readying to receive a charge or setting up tripwires.
In your case the Dodge skill would represent his ability to react to those area attacks in an appropriate manner. If he's there at the moment of the attack, that's where he is. The attacker has been leading him with a rocket launcher or he's just tripped a mine or whatever.
Why this makes sense
The logical reasoning behind this is that you have the ability to run very fast but it doesn't translate into making your arms move incredibly fast or increase the number of attacks you can have. For an example, check out the landlady chase scene in Kung Fu Hustle (which you should have watched as source material anyway :P).
For you
If you're in open spaces, don't worry about staying on the map. Hit and run tactics are what you've chosen to focus on.
For your GM
It isn't a problem that one of your players doesn't have to be on the map to attack people. He's constantly at long range. So what. It just means that to defeat him, people are going to have to find some interesting counters to his tactics. Counterattacks, land mines, slowing auras, tight quarters and so on.
Those will validate his choices and add challenge to the game.
Given that you appear to have all the rules already, it seems that the problem is in interpretation. Given that, my view is :
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
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.
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.
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.
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.
Best Answer
Assuming speed 6:
A square is 5'. Running gives him two extra squares of movement per move action, so he'll move 8 squares per action. He gets three actions in your scenario; thus, he's moved 24 squares. 24 squares is 120 feet.
A round is six seconds. Therefore, the human is moving 120 feet in six seconds, or 20 feet per second. That's 72,000 feet per hour, or about 13.6 miles per hour, although he can't sustain that speed for more than one round.