[RPG] Does moving a grappled opponent while mounted reduce your mount’s speed

Shadow Step only teleports you, so you can't bring anyone else (willingly or unwillingly) with you. Compare with other ways to teleport like the spell teleport (PHB, p. 281) which only allows bringing others with you because it explicitly says so.

If you could somehow bring a grappled creature with you through a Shadow Step (a custom Feat, perhaps?), it would not reduce the distance. Shadow Step's distance limit is a fixed 60 feet that is independent of your speed statistic, and drag-grappling only reduces your speed.

On the plus side, the combination of these two points makes Shadow Step a convenient way to escape a grapple against you.

No, you can not.

The grappling rules state that you can move a target you are grappling:

Moving a Grappled Creature. When you move, you can drag or carry the grappled creature with you, but your speed is halved, unless the creature is two or more sizes smaller than you.

But, this applies only to you and your movement.

In D&D 5e, your movement is completely separate from the mount's movement. It even has its own turn distinct from yours; you can only control where it moves. Per the rules on controlling a mount:

The initiative of a controlled mount changes to match yours when you mount it. It moves as you direct it, and it has only three action options: Dash, Disengage, and Dodge.

Thus, the mount's movement will not allow you to move the grappled creature, and counts as an effect that removes the grappled creature from your reach:

The condition also ends if an effect removes the grappled creature from the reach of the grappler or grappling effect, such as when a creature is hurled away by the thunderwave spell.