Should I TM my Magikarp?

No, just evolve him to Gyarados.

Well, even with Magikarp's Revenge, his abysmal stats still make him subpar. Vaporeon is definitely a better choice for a water attacker, most people even refuse to acknowledge Magikarp having this option.

Even if you want to, it is impossible for now. Magikarp can only learn Splash and Struggle.

I want my Magikarp to be special.

The "battles won" number on your pokemon includes the number of defending pokemon defeated while your pokemon was one of your selected six attackers, even if it never actually participated in the fight. I rarely need more than three attackers, so I have been selecting a Magikarp as my sixth pokemon to build up a crazy high "battles won" number on it.

Why not? :-D

(Type of sensible answer but gets downvoted):

Hell no.

(Type of answer folks want to see when they ask these questions now a days);

Oh hell yes! He is a water only type which means he has no double weaknesses and can be a good defender unlike the old big 7! Also, he resists water and ice so he is one of the few mons that does not have a weakness to one of Lapras' movesets and that doesn't happen often in this current meta! Note the fact that his charge move is normal so you will find it generally unresisted by most mons! To add to that, see the fact that his defensive stat is almost 4X his attack stat and, buddy, you've got yourself a defender! He resists steel moves so he should be great attacking into Steelix, the best defender in the game. Power that sucker up!

Diversity!!!