[RPG] Artificer Armorer Guardian attack and cantrips like Booming blade

Lightning Launcher cannot be used with two-weapon fighting

The requirement to use two-weapon fighting is to be holding 2 light melee weapons, one in each hand. The Lightning Launcher is neither light, nor a melee weapon, nor handheld, so it does not qualify for two-weapon fighting.

You can still attack with the launcher and another weapon with Extra Attack

However, as a Armorer of 5th level or higher, you have Extra Attack, and there's nothing to stop you from using your 2 attacks to attack with 2 different weapons. So in the situation you describe, where the armorer wields a shield in one hand and a weapon in the other, and also has the lightning launcher equipped, they could make one attack with the lightning launcher and the other with the weapon in their hand, all while keeping their shield equipped.

(Of course, you can also just attack twice with the launcher, or with your held weapon.)

Your armor's gauntlets don't occupy your hands while worn

While in use, armor is worn, not carried. So the gauntlets that are part of the armor you are wearing don't occupy your hands, even while they "count as simple melee weapons". This means that even though you have a weapon available on each hand, your hands are still free to perform the somatic components of spells or perform any other action. (This versatility is one of the benefits of the Arcane Armor feature.)

You can use your arcane armor as a focus while wearing it

Normally, you need to (be able to) hold the material component of a spell in your hand in order to cast it:

A spellcaster must have a hand free to access a spell's material components -- or to hold a spellcasting focus -- but it can be the same hand that he or she uses to perform somatic components.

So you might be worried that you have to take off your armor and carry it in order to cast spells with it. However, the Arcane Armor feature specifically overrides this general rule, because the ability to use it as a spellcasting focus appears in the list of benefits you gain while wearing the armor, not carrying it:

You gain the following benefits while wearing this armor:

• [...]
• You can use the arcane armor as a spellcasting focus for your artificer spells.
• [...]

In fact, since you gain these benefits only while wearing the armor, this means that you actually cannot use your armor as a spellcasting focus while holding it. Contrast this with infusing a suit of armor with Enhanced Defense, which would technically allow the artificer to use the armor as a focus while holding it rather than wearing it. (Of course, if you infuse a suit of armor with Enhanced Defense and then also use your action to turn it into your arcane armor, then you would be able to use the armor as a spellcasting focus while either holding or wearing it.)