How do these feats work together?
Shot On The Run is a full-round action, whereas Rapid Shot and Many Shot both require a full-attack action. Thus you cannot use either with Shot On The Run.
However I don't see anything to say that you can't use Rapid and Many shot together. So assuming a +6 BAB, you could attack with +4/+4/-1, with the first +4 attack using two arrows (for a total of 4; that'll run through a quiver right quick).
Shot on the Run (Combat) - final
You can move, fire a ranged weapon, and move again before your foes can react.
Prerequisites: Dex 13, Dodge, Mobility, Point-Blank Shot, base attack bonus +4.
Benefit: As a full-round action, you can move up to your speed and make a single ranged attack at any point during your movement.
Normal: You cannot move before and after an attack with a ranged weapon.
Rapid Shot (Combat)
You can make an additional ranged attack.
Prerequisites: Dex 13, Point-Blank Shot.
Benefit: When making a full-attack action with a ranged weapon, you can fire one additional time this round. All of your attack rolls take a –2 penalty when using Rapid Shot.
Manyshot (Combat)
You can fire multiple arrows at a single target.
Prerequisites: Dex 17, Point-Blank Shot, Rapid Shot, base attack bonus +6.
Benefit: When making a full-attack action with a bow, your first attack fires two arrows. If the attack hits, both arrows hit. Apply precision-based damage (such as sneak attack) and critical hit damage only once for this attack. Damage bonuses from using a composite bow with a high Strength bonus apply to each arrow, as do other damage bonuses, such as a ranger's favored enemy bonus. Damage reduction and resistances apply separately to each arrow
The Shot on the Run description states a "single ranged attack". Rapid Shot and Manyshot both count as full-attack actions which only allow for a 5 foot step. So by the rules they don't work together.
I was actually going to say that about combining Manyshot with Rapid Shot as well, considering Manyshot is counted as a single attack. Just no movement beyond a 5ft step.