Clarifying some misconceptions
It is my understanding that you and your mount basically share a move action but not necessarily a standard action.
Incorrect. You move with your mount (on your mount’s move action), but it does not consume your move action. There are limited options for what you can do with that move action, but you still have it. That’s important, because you need it for full-round actions.
Note that mounts that are not war-trained require move actions to control them during combat.
[Not sharing a standard action] allows you to make full round attacks even if you mount takes a double move, but a penalty is applied to your attack roll.
Full-round actions require both a move and standard action; if you use your move action to control your mount, you only have a standard and thus can only make an attack, not a full-attack.
But for war-trained mounts, animal companion mounts, and others, this is more-or-less correct, because you have both move and standard available. You still cannot make a melee full-attack, however:
If your mount moves more than 5 feet, you can only make a single melee attack. Essentially, you have to wait until the mount gets to your enemy before attacking, so you can’t make a full attack.
However, there are no penalties for melee:
Even at your mount’s full speed, you don’t take any penalty on melee attacks while mounted.
Unless your mount charges, in which case it counts the same as if you were charging. Also, double damage with lances:
If your mount charges, you also take the AC penalty associated with a charge. If you make an attack at the end of the charge, you receive the bonus gained from the charge. When charging on horseback, you deal double damage with a lance.
Note also that if you have the Pounce ability, you could make a full-attack at the end of a mounted charge: you get the bonus gained from the charge (usually +2 to attacks, but with Pounce, a full-attack), and the “issue” that you can only attack after moving is the norm for Pounce.
On the other hand, ranged attacks are allowed to take full-attacks:
You can make a full attack with a ranged weapon while your mount is moving.
But there are penalties if your mount moves too much:
You can use ranged weapons while your mount is taking a double move, but at a -4 penalty on the attack roll. You can use ranged weapons while your mount is running (quadruple speed), at a -8 penalty.
Now, your question:
Could I shoot a target during my mounts single initial move action to approach an enemy,
Yes, you are allowed to make ranged attacks while your mount is moving. It can be a full-attack if you did not use your move action controlling the mount.
In either case [i.e. mount double-moves or runs], you make the attack roll when your mount has completed half its movement.
This rule suggests that the halfway point of the move is the usual place to adjudicate these, though technically it applies only to the double-move or run situations. There is no rule that I can find about normal movement and where shots come from; I’d allow it from wherever and the statement that “you can make a full attack with a ranged weapon while your mount is moving,” certainly implies that it’s during the movement, i.e. somewhere in the middle, but check with your DM.
and then do nothing while my mount attacks the target for its standard action?
Yes, if your mount still has its standard action, it can use it to attack after its move. Note that double-moves, runs, and charges are all full-round actions that do not leave the mount a standard action in which to attack (though a charge involves an attack to begin with, or more than one if your mount has Pounce).
I would still need to apply the penalties from shooting from a moving mount
Only if the mount double-moves or runs, but if it does either of those (both are full-round actions), it would not have a standard action in which to attack at the end.
roll for fight with warhorse
this would allow me to avoid attacks of opportunity.