So as long as you qualify for Sneak Attack, it applies. In your example, the melee attack would trigger Sneak Attack.

You can even get sneak attack on an opportunity attack later in the round because it's only limited to once per turn. Sneak attack is not an attack action, it's a triggered event based on its prerequisites.

The official Sage Advice Compendium confirms this:

Can you use green-flame blade and booming blade with Extra Attack, opportunity attacks, Sneak Attack, and other weapon attack options?

Introduced in the Sword Coast Adventurer’s Guide, the green-flame blade and booming blade spells pose a number of questions, because they each do something unusual: require you to make a melee attack with a weapon as part of the spell’s casting. First, each of these spells involves a normal melee weapon attack, not a spell attack, so you use whatever ability modifier you normally use with the weapon. (A spell tells you if it includes a spell attack, and neither of these spells do.) For example, if you use a longsword with green-flame blade, you use your Strength modifier for the weapon’s attack and damage rolls.

Second, neither green-flame blade nor booming blade works with Extra Attack or any other feature that requires the Attack action. Like other spells, these cantrips require the Cast a Spell action, not the Attack action, and they can’t be used to make an opportunity attack, unless a special feature allows you to do so.

Third, these weapon attacks work with Sneak Attack if they fulfill the normal requirements for that feature. For example, if you have the Sneak Attack feature and cast greenflame blade with a finesse weapon, you can deal Sneak Attack damage to the target of the weapon attack if you have advantage on the attack roll and hit.

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.