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If someone with frag grenades assaults a unit with defensive grenades AND is in cover, am I interpreting the rule book correctly by saying that the defender still gets to go first since they are in cover and the grenades cancel each other out?

Assault Grenades don't cancel Defensive Grenades.

Assault cancels the initiative penalty for cover.
Defensive cancels the extra attack from a charge.

As above. there is no way to read them such that they cancel out, as they deal with entirely different effects in game.

Defender gets no initiative boost due to Assault grenades. Attacker gets no charge from the defence grenades.

(Although unsure if the defensive grenades cancel other benefits of charge such as Furious Charge, will need to check the rulebook when I get home)

Youre also thinking 4th edition. In 5th edition a unit assaulting through cover drops their I to 1, the defender does not gain a bonus.

No, defensive grenades have absolutely no effect on anything other than the assault bonus

Yeah, furious charge isn't affected.