If my Fighter (Eldritch Knight) use War Magic (PHB p. 75) to use the cantrip Booming Blade, will I be able to use my Extra Attack, and bonus action attack, to make 3 attacks at level 7?
No you cannot use Extra Attack with Booming Blade
All definitions of Extra Attack say something like (Fighter text quoted):
Beginning at 5th level, you can attack twice, instead of once, whenever you take the Attack action on your turn.
Booming Blade is a spell and it requires you to take the Cast a Spell action - the Cast a Spell action is not the Attack action.
The reason you can attack with your weapon while taking the Cast a Spell action is that it is part of the Booming Blade spell:
As part of the action used to cast this spell, you must make a melee attack with a weapon against one creature.
Yes, you can use your Bonus Action to attack with War Magic
when you use your action to cast a cantrip, you can make one weapon attack as a bonus action.
There are no limitations on they type of action you must take or on the effects the cantrip has.
No. You will make one attack as part of the Booming Blade cantrip, and one attack as a bonus action.
Extra attack requires you to take the Attack action in order to gain the additional attack(s) since neither making an attack as part of the Cast a Spell action nor making on as a bonus action are part of the Attack action you do not get to make any additional attacks with it.
MAKING AN ATTACK
Whether you're striking with a melee weapon, firing a weapon at range, or making an attack roll as part of a spell, an attack has a simple structure.
Choose a target.
Pick a target within your attack's range: a creature, an object, ar a location.
The DM determines whether the target has cover and whether you have advantage ar disadvantage against the target. In addition, spel1s, special abilities. and other effects can apply penalties ar bonuses to your attack roll.
Resolve the attack.
You make the attack roll. On a hit, you rol1damage, unless the particular attack has rules that specify otherwise. Some attacks cause special effects in addition to ar instead of damage.
If there's ever any question whether something you're doing counts as an attack, the rule is simple: if you're making an attack rol1,you're making an attack."
The last line of the rule says it all, if the spell requires you to make an attack roll (not a spellcasting roll) it counts as an attack.
So you CAN extra attack after using booming blade. You can even combo several Booming blade cantrips if you have extra attacks allowing you to do so.