Yes.
The actual rules for this are oddly worded, but the notation is pretty simple.
In general, there are two formats for "things that increase your AC." The most common format is the one attached to armor:
[your AC equals] 11 + Dex modifier
or Draconic Resilience:
your AC equals 13 + your Dexterity modifier.
Note that there is no "plus" at the beginning of the formula. These items set your AC to a specific value.
The other format is that used by shields and class features that increase your AC above its base value:
Defensive fighting style:
You gain a +1 bonus to AC.
Or a shield:
Wielding a shield increases your AC by 2.
These are phrased in terms of increasing, rather than setting your AC. They are preceded by the phrase "increases" or a plus sign.
So, it's pretty simple. You get one thing that sets your AC, and any number of things that modify it (except where restricted by other rules, like the rules stating you can only use one shield).
A fighter could very easily have an AC like this:
[15 + Dex Mod] (half-plate) [+2] (shield) [+1] (defensive fighting style)
For a total AC of 18 + Dex Mod.
But he couldn't add a chain shirt to increase his AC, because both that and the half-plate set his AC to a fixed value.
The same applies to Draconic Resilience. You can stack it with a shield, because a shield increases AC. And you can further stack on the Shield spell, giving you a (temporary) total of [13 + Dex Mod] [+2] [+5], or 20 + Dex Mod.
But you can't combine Draconic Resilience with Mage Armor, because both Mage Armor and Draconic Resilience are trying to set your AC to a fixed value.
Some things Draconic Resilience stacks with:
Some things it does not:
RAW
The rules for AC are on page 14 of the Player's Handbook, or page 9 of the Basic Rules.
The relevant part is this:
Without armor or a shield, your character’s AC equals 10 + his or her Dexterity modifier. If your character wears armor, carries a shield, or both, calculate your AC using the rules in chapter 5. Record your AC on your character sheet.
[...]
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.
The rules called out above appear to be these:
Wielding a shield increases your Armor Class by 2. You can benefit from only one shield at a time.
If you wear light armor, you add your Dexterity modifier to the base number from your armor type to determine your Armor Class.
(and so on, for each type of armor)
The rules are sloppy here. Technically, they don't give you a calculation for AC when wielding a shield and no armor. And they never explicitly call out what is or is not a "way to calculate your AC."
With that said, we can use examples to build the intent of the designers.
It's pretty clear that you don't have to choose between a shield and plate. It's also pretty clear that a shield shouldn't set your AC to 2.
Finally, it's pretty clear that wearing a chain shirt with half plate shouldn't set your AC to 28 + 2 * Dex modifier (max 2).
It seems obvious then, that a "calculation" is something that sets your AC to a value plus other modifiers.
If something just adds on to (or increases) your AC, you continue to use whichever calculation is most favorable plus the new modifier.
Yes
A shield is an item. You can pick up an item without an action (Player's Handbook p.190).
Wielding a shield, not merely holding it, increases your Armor Class by 2 (Player's Handbook p.144). Donning a shield takes an action. (PHB p.146)
PHB p.146 says that armor must be donned to benefit from it.
If we wish to be exceptionally literal, p.144 says that armor and shields are separate things. A shield is carried in one hand. It must be wielded, and the dictionary definition of wield is "To handle with skill and ease, especially a weapon or tool". Given the descriptive text on p.144 that "Anyone can put on a suit of armor or strap a shield to an arm", we can most probably assume that shields are strapped to an arm, and that this is what takes an action, and that to don a shield in its proper fashion is necessary in order to handle it "with skill and ease". I think one could make a valiant stand to rules-lawyer that donning a shield isn't necessary doesn't do anything, but the meaning seems clear.
Either way, to don is to wear. You can pick up, hold or carry a shield without going to the additional effort of wearing it. No rule forbids you from merely picking up the shield the way you would pick up any other dropped item.
Best Answer
It’s your shield, so it’s up to you.
There is no official “direct” answer to your question, except possibly what’s found in the introduction to the Player’s Handbook, in “How to Play the Game”:
As the player, you decide what you character does. If that means you wear your shield all the time, then you wear your shield all the time.
As far as sleeping with your shield on, realistically, adventurers probably don’t do that; so talk to your DM about that one. In my games, as a player, I don’t sleep with my shield, and as a DM, I assume my players don’t either. This really only matters if the party is ambushed while sleeping. When this happens, it can help raise the stakes a bit on that first round, as shield users are forced to decide between their shield AC bonus and using their action for something else.