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, Fey Ancestry would grant advantage

Jeremy Crawford provides a clear answer to this question here:

Does "Fey Ancestry" apply to any spell or spell-like ability that gives the "charmed" condition?

Fey Ancestry benefits any saving throw meant to stop an elf from suffering the charmed condition.

From his wording, it's clear that Fey Ancestry would benefit the half-elf character in your case. The advantage on saving throws is not merely on saves that, if failed, would impose a new charmed condition; it applies to any saving throw that, if successful, would result in the character not being charmed. This would include saving throws made to break free from a charm effect.