DC on Mia?

You CAN, but I would advise against it. Mia's defense stat isn't necessarily great, and her resistance is even worse, so, in my opinion, putting Distant Counter on her is not something I'd do, especially when there are A skills that can better increase her Player Phase abilities rather than trying to make her something she's not really suited out to be.

DC is classically a defensive skill. Used to bait enemies. Either that, or mixed phase.

The point here is: Mia does not only lose her best ability, Flashing Blade, but also gets a skill which fits a totally different playstyle, which is usually not good.
It can be, when you rework her build, but as you say, you like her as a unit, I don't think you really want to do that.

DC is usually best for heroes with a balanced defensive statline, or at least one of the defensive stats being very high, so that this unit can counter more enemies, which are not able to pierce his bulk. At this point you might consider common ranged threats, like Brave Bow Lyn, Reinhardt, Nino and Celica. Look for the ones which hurt you the most, then look for a hero who can counter them.

DC is a very misleading skill to copy sets from other players. It is has the highest SP costs from all A-Slot abilities, so in the highest arena Tier it is like mandatory to use this skill. Or at least very common. That's why a lot of people up there use this skill on many heroes. Sometimes even heroes, who could be better with another skill, but this skill is not as expensive, so they choose DC instead to get more points. For now, you should not aim for this. Those players have much more Hector than you and me.

L!Ike is a better tank because it's his job to be a tank. Mia's job is not to be a tank. Her job is to be an offensive attacker.

First of all NO! It takes away the sweet advantage of flashing blade. She isn't a unit that is defensive and could get 1 shot by 2 ranged foes like Reinhart or Male Morgan. It would be best to keep flashing blade since she is a high speed unit.