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.

You can't add Dueling damage to Shocking Grasp but you could add it to the initial attack from Booming Blade

The Dueling fighting style says:

When you are wielding a melee weapon in one hand and no other weapons, you gain a +2 bonus to damage rolls with that weapon.

So, the bonus damage from Dueling applies only when you are making damage rolls with a weapon held in one hand (and not holding another weapon). Do these spells meet that criteria?

Shocking Grasp:

Lightning springs from your hand to deliver a shock to a creature you try to touch. Make a melee spell attack against the target.

This is a melee spell attack with a range of 'touch' and there is no mention that the attack is made via any sort of weapon. You aren't making a weapon damage roll, so you don't gain the Dueling damage bonus.

Your damage for Shocking Grasp is thus the normal 1d8 (unless you have any other reason to add damage to this roll).

Booming Blade:

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, otherwise the spell fails. On a hit, the target suffers the attack’s normal effects, and[...]

In this instance you are making a 'melee attack with a weapon' and the target suffers 'the attacks normal effects'. If you have taken the Dueling fighting style (and this attack meets the usual criteria - wielded in one hand, while holding no other weapons) then the 'normal effect' of an attack includes your bonus Dueling damage as part of the damage roll.

Your damage for the initial attack from Booming Blade is thus:
your weapon damage dice (1d8) + your ability modifier (probably strength or dex) + Bonus Dueling damage (2)

Subsequently Booming Blade may deal additional thunder damage but the Dueling bonus is only added once.

When / if the additional thunder damage occurs, it comes from the spell and not the weapon. The bonus that Dueling adds is to damage the weapon deals; the weapon attack is part of the spell, but the secondary damage instance (the thunder damage) isn't from the weapon.

[...] If the target willingly moves before then, it immediately takes 1d8 thunder damage, and the spell ends.