[RPG] Do natural armor bonuses stack if they’re from different sources

They don't, but they don't really need to.

Generally speaking, most bonuses of the same type do not stack. Instead, only the highest bonus applies.

So your Natural Armor bonus will not stack with that of your Eidolon, since both bonuses are of the same type, and there is no exception to the general rule found for the Fused Eidolon ability.

However, look at the following parts

Fused Eidolon

[...] The synthesist uses the eidolon’s base attack bonus, and gains the eidolon’s armor and natural armor bonuses and modifiers to ability scores. [...]

While fused, the synthesist loses the benefits of his armor.

(emphasis mine) Note that you do not lose your Natural Armor while fused (because the rules don't say so)

Armor Bonus
The number noted here is the eidolon’s base total armor bonus. This bonus may be split between an armor bonus and a natural armor bonus, as decided by the summoner. [...]

So instead of taking your Eidolons Natural Armor Bonuses, set them to zero. Put all your Eidolons Armor Bonus to regular armor and keep your own Natural Armor.

Sort of, but no.

Unarmored Defense does not provide a bonus to AC. It provides an alternate means of calculating it.

So, your druid has two AC calculations available to him:

1. AC provided by beast form.

2. 10 + Dexterity Modifier + Wisdom Modifier

You don't get to add the two together, because they both set AC to an absolute value. A brown bear has an AC of 11. It does not have a +1 natural armor bonus.

Likewise, a Monk using unarmored defense has an AC of 10 + Dexterity Modifier + Wisdom Modifier. He does not simply add Wisdom to his AC.

Your Druid can use Unarmored Defense in beast form, but he has to choose between that and the fixed AC of the form. For a brown bear, his options are:

1. 11

2. 10 + 0 (Dex) + Wisdom Modifier

The Unarmored Defense will probably give him better AC than the bear had, but less than a typical monk has (due to the low dexterity).

For more detail on 5e's armor notation, see this answer.

Natural Armor

Instead of using a flat AC of 11, you can use the bear's natural armor calculation. This allows you to calculate the bear's AC in cases where its dexterity or other stats change.

The armor calculation for natural armor can be found in the January 2016 rules answers:

Natural Armor: 10 + your Dexterity modifier + your natural armor bonus. This is a calculation method typically used only by monsters and NPCs, although it is also relevant to a druid or another character who assumes a form that has natural armor.

[...]

Similarly, a druid/barbarian who transforms into a beast form that has natural armor can use either the beast’s natural armor or Unarmored Defense [...].

In this case, you have the choice of either of two calculations:

1. 10 + Dexterity Modifier + Wisdom Modifier (Unarmored Defense)

2. 10 + Dexterity Modifier + Natural Armor Modifier (Natural Armor)

You don't get to mix and match between these two formulas; you have to pick one.

Thanks to Aviose for finding this.

See Also

Jeremy Crawford has tweeted about Unarmored Defense and Wildshape, stating that you must choose either the beast's AC calculation or the Unarmored Defense calculation.