Pokemon-go – Will a Pokemon take it’s type into account when attacking with moves of a different type

Yes, it does. But only at 1.25x. Here is the decoded protobuf file. In it is this:

Items {
  TemplateId: "BATTLE_SETTINGS"
  BattleSettings {
    RetargetSeconds: 0.5
    EnemyAttackInterval: 1.5
    AttackServerInterval: 5
    RoundDurationSeconds: 99
    BonusTimePerAllySeconds: 10
    MaximumAttackersPerBattle: 20
    SameTypeAttackBonusMultiplier: 1.25
    MaximumEnergy: 100
    EnergyDeltaPerHealthLost: 0.5
    DodgeDurationMs: 500
    MinimumPlayerLevel: 5
    SwapDurationMs: 1000
  }
}

The damage isn't normalized I believe - the only proof I have is that it wasn't ever stated otherwise and therefore implicitly stating it.

According to the Pokemon Go Database, Fury Cutter has a damage of 3 with an attack speed of 0.4 while Confusion has a damage of 12 with an attack speed of 1.51. This again means Confusion's DAS (Defense Attack Speed) is 3.01 while Fury Cutter's DAS is 1.9. By dividing the damage with the DAS, we conclude that Fury Cuter has a Defense DPS of ~1.58 while Confusion has ~3.99.

This would suggest that an attack that usually has a 5,3% higher dps actually has a ~152,5% higher dps when defending.

While those certainly are 2 extreme examples, it shows that you should use Pokémon with high Base Damage over Pokémon with high DPS to defend your Gym.

I hope I answered this to your satisfaction ~Crowley