No, Rage is only replaced when you enter the Supernatural Fury

The way this is worded, the Rage feature is not removed upon reaching level 3. Instead the feature is disabled while in the Supernatural Fury. This means that you could enter a "regular rage" but these features are disabled while in the alternate state.

This shows that the wording is probably flawed

Because of the earlier bit in the feature: "You gain the ability to cast and concentrate on spells, even while raging", you could also cast spells while in a "regular rage." This is almost certainly not the intent.

Yes, it will last the full minute*

*Unless the effect is removed by some other mechanic.

There is a difference between Casting a Spell (see Chapter 11 of PHB) and gaining the effects of a spell as if you had cast it. In this case, the effect is external.

The wand is providing the effects of the Reduce/Enlarge spell but doing so without having you cast it. Since you aren't casting it, you also have no control over how long it lasts.

Can you turn it off at will?

It's not a concentration spell that you have control over so RAW it doesn't seem like you can turn it off. This is similar to ending a non-concentration spell early and that RAW it doesn't seem possible but I think many tables allow it. In this case, you are controlling the wand and it wouldn't be unreasonable for a DM to allow you to turn the effect off at will. However, that would be on a DM decision and not purely within the mechanics.

Removable via other mechanics/abilities

This is still a magical effect and a spell effect. Mechanics such as dispel magic or even a Paladin's cleansing touch could remove the spell effect early if a DM didn't permit the wand owner to do so (or if someone else wanted to end it.)