What’s the amount of interaction needed to use a spellcasting focus? Does it have to be included in the somatic component?
If a spell has a material component, you need to handle that component when you cast the spell (PH, 203). The same rule applies if you’re using a spellcasting focus as the material component.
If a spell has a somatic component, you can use the hand that performs the somatic component to also handle the material component. For example, a wizard who uses an orb as a spellcasting focus could hold a quarterstaff in one hand and the orb in the other, and he could cast lightning bolt by using the orb as the spell’s material component and the orb hand to perform the spell’s somatic component.
Another example: a cleric’s holy symbol is emblazoned on her shield. She likes to wade into melee combat with a mace in one hand and a shield in the other. She uses the holy symbol as her spellcasting focus, so she needs to have the shield in hand when she casts a cleric spell that has a material component. If the spell, such as aid, also has a somatic component, she can perform that component with the shield hand and keep holding the mace in the other.
If the same cleric casts cure wounds, she needs to put the mace or the shield away, because that spell doesn’t have a material component but does have a somatic component. She’s going to need a free hand to make the spell’s gestures. If she had the War Caster feat, she could ignore this restriction.
It depends on the VSM components of the spell. With hands full you can still cast any Verbal-only spell, or anything with no components at all, like some racial feature spells (Gith psionics). Without the War Caster feat the EK would need to sheathe the sword (using their object interaction) or drop it to cast anything else. The main disadvantage of this is that they cannot make an attack of opportunity with the weapon that turn, and must use a future object interaction to draw the sword. With the War Caster feat they would be able to cast any V, S or VS spells while still holding the weapon and shield. If they have a way of using a weapon as a focus (it's a staff, Ruby of the War Mage, Bard of Swords, Warlock Improved Pact Weapon), or if they are a cleric/paladin with their holy shield, then they can cast V, M, VM and VSM spells while armed. With a weapon or shield that is a focus plus the War Caster feat you are able to cast any spell while fully armed.
If say an eldritch knight that wants to cast a spell, but has a sword and shield in his hands, does he need to put one down to cast the spell?
This is covered in Sage Advice.
The Forum Infestation (TM)
It depends on the VSM components of the spell. With hands full you can still cast any Verbal-only spell, or anything with no components at all, like some racial feature spells (Gith psionics). Without the War Caster feat the EK would need to sheathe the sword (using their object interaction) or drop it to cast anything else. The main disadvantage of this is that they cannot make an attack of opportunity with the weapon that turn, and must use a future object interaction to draw the sword. With the War Caster feat they would be able to cast any V, S or VS spells while still holding the weapon and shield. If they have a way of using a weapon as a focus (it's a staff, Ruby of the War Mage, Bard of Swords, Warlock Improved Pact Weapon), or if they are a cleric/paladin with their holy shield, then they can cast V, M, VM and VSM spells while armed. With a weapon or shield that is a focus plus the War Caster feat you are able to cast any spell while fully armed.
Simple, no?