If a multi class Paladin/ Eldritch Knight cast Shield of Faith, and then as a reaction cast shield, would they get a +7 Bonus to their AC?
All bonuses to AC stack.
I assume you're thinking of previous versions of D&D, where bonuses of the same type didn't stack. However, in 5e, the majority of ways to gain AC are separate ways to calculate your base AC, such as Plate Armour, which sets your base AC to 18, or Mage Armour, which sets your base AC to 13 + Dex. From the Player's Basic Rules, page 9:
Some spells and class features give you a different way to calculate your AC. If you have multiple features that give you different ways to calculate your AC, you choose which one to use.
Actual bonuses to AC are very rare in 5e, and stack fully with each other and with different sources of AC, unless the source or the bonus specifies otherwise. An example of this is the Monk's Unarmoured Defense ability, which specifies that it only applies when you are not carrying a shield. Note that it says nothing about the Shield spell, so that would function normally.
An example of non-stacking bonuses is the physical shield itself, which specifies that you can benefit from only one shield at a time. So carrying 2 shields only gets you a +2 bonus. (Player's Basic Rules, pg 44.) Note that the Shield spell provides you with 'an invisible barrier of magical force', not a (lowercase) shield. (Player's Basic Rules, pg 100.) So it does not come under this exception and stacks with a physical shield without difficulty.
So a shield stacks with the Shield spell, and both of them stack with any armour you are wearing, or Mage Armour if you have that instead.
You can't cast spells while holding your two-handed sword or your sword and shield without the War Caster feat. This is spelled out on page 203 of the PHB:
If a spell requires a somatic component, the caster must have free use of at least one hand to perform these gestures.
However, you don't necessarily need to take the feat in order to use a two-handed sword or a sword and shield and cast spells. (I realise the previous sentence seems to contradict the first one, but I'm getting there.)
Each turn, you can interact with one object without using an action. (See page 190 of the PHB for details.) If you are using a sword and shield, this means you can, for example, sheathe your sword without using an action, then cast a spell on the same turn. Then next turn, when you want to attack with your sword, you can draw it without using an action, then attack on the same turn. Note that this leaves you unable to take opportunity attacks between these 2 turns.