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b) Official rule reading (by many for years) is that YOU CAN attack only with your Bonus Action offhand (but do not add Ability modifier unless you have a special feature exception). This logic does not apply to Monk's extra unarmed attack for which you must first do a normal Action attack (Monks are not in the game, but just saying for your D&D knowledge).
EDIT : I was wrong or rather the popular discussions about this on the web were wrong. You must first do the attack Action, before using your offhand.
Agreed on a) but you're wrong on b).
General Rule: You can only make a Bonus Action attack if you used your Action to make an Attack.
Specific Rule: Berserker Barbarian, etc.
My post right after states that Zress was correct and I was wrong. 