No, You cannot combine attacks with booming blade.
Both Booming Blade and Green Flame Blade say the following:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature within the spell's range
This means that these cantrips do not require you to take the Attack action, they require you to take the Cast a Spell action.
If we look at extra attack, it is defined as:
Beginning at 5th level, you can Attack twice, instead of once, whenever you take the Attack action on Your Turn.
No Attack action is taken, so no extra attacks can be used. Additionally, two weapon fighting states:
When you take the Attack action and attack with a light melee weapon that you’re holding in one hand, you can use a bonus action to attack with a different light melee weapon that you’re holding in the other hand.
Again, no attack action is taken, so you cant use a bonus action to make an off hand attack.
In your scenario, you can cast hunter's mark or hex, and then the hit from the booming blade will trigger those effects, as well as sneak attack (if the conditions are met), but you will be unable to make more attacks after that.
Action surge gives you another whole action, which you can use to cast a spell, like Booming Blade which will give you one more attack, or if you just use a regular attack action, extra attack and two weapon fighting could apply.
Just a different instance of a weapon
Weapons can be of the same type (ie two short swords), but you must have two actual weapons to use two weapon fighting.
The wording is only there to prevent someone from trying to use two-weapon fighting with one weapon.
The dual-wielder feat does supply circumstantial evidence for this when it says:
You can use two-weapon fighting even when the one handed melee weapons you are wielding aren’t light.
But really it is just common sense in this case. The wording would have been much more specific (perhaps "...with a light melee weapon with a different name") if the kind of restriction you propose was the intended reading.
Best Answer
The quote you've provided is pretty clear. Hunter's Mark deals an extra 1d6 damage whenever you hit the target with a weapon attack. So the off-hand attack will deal an extra 1d6, as will any additional attacks from Extra Attack, or any opportunity attacks you make on the target.