[RPG] Does Wand of the War Mage bonus apply to spells like GFB and Booming Blade

The bonus applies to all spell attack rolls, whether the source is from a magic item or a class spell.

Wand of the War Mage says:

you gain a bonus to spell attack rolls...

It doesn't say anything about the spell even having to be cast or that the bonus only applies to class spells unlike something such as the Rod of the Pact Keeper which specifically states:

you gain a bonus to spell attack rolls and to the saving throw DCs of your warlock spells.

It doesn't matter if you're using an object or casting a spell, the bonus still applies.

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.