Riding the shield? Stabbing with the arrow, then firing it? Impaling two enemies with one shot? Climbing and shooting a moving foe? Being an elf who is sufferable?
Almost everything there is doable:
1. Riding a shield: This is most likely declare an action although many DMs would rule you don't need an action, just like if it were a sled or carriage you were on.
2.a. Stabbing with an arrow: improvised weapon attack
2.b. Then firing it: ranged weapon attack
3. Impaling 2 enemies with one shot: Arcane Archer Piercing Arrow (I admit I had to think about this one for a while)
4. Climbing on a moving foe and attacking it: Optional rule from DMG page 271. In terms of fiction, the scene from LOTR is what I think of when I read this section of rules. Also note RAW, not only can you climb on and attack the creature, you have advantage on those attacks. I have done this a lot in game with a variety of characters. I've done it so much one of the DMs I play with nerfed this and houseruled it so it no longer gives the advantage.
5. Being an Elf who is sufferable: Ok you have me there!