You can wield & attack with two weapons without any special class feature or feat.
Like most contests, however, some restrictions may apply. From the PHB, p195 (emphasis mine):
Two-Weapon Fighting
When you take the Attack action and attack with a light melee weapon that you're holding in one hand, you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand. You don't add your ability modifier to the damage of the bonus attack, unless that modifier is negative.
As long as both weapons are light weapons, you can attack with both at the cost of an action for the first attack, as normal, and then a bonus action for the second attack. The second attack does not receive your ability modifier as a bonus to damage the way attacks normally would.
Fighters & some other classes have an option for their Fighting Style class feature which is also called, confusingly, Two-Weapon Fighting. This feature is not required to attack with two weapons as described above; the benefit it gives is to allow you to add your ability modifier to the second attack's damage.
The Dual Wielder feat is likewise not required to attack with two weapons as described above; it instead provides additional benefits and removes other restrictions/penalties when you choose to do so.
Dual Wielder and the Two-Weapon Fighting style make you better at fighting with a weapon in each hand, but are not required to do so.
No, and Yes.
First of we need to consider this post by Jeremy Crawford:
Dual Wielder is intended to work with actual weapons. Using the feat with improvised weapons is up to the DM.
See also: Shield and Improvised Weapon
It is not RAI to use a shield or other improvised weapon with the Dual Wielder feat. The rest that follows assumes you talked your DM into allowing that.
One Bonus Action
can I attack 4x in my Attack action (axe, shield, axe, shield)?
You can attack twice in your attack action, after level 5.
You, however, only get one bonus action to make one more attack if you attack with a light weapon you could attack (axe, axe, Shield), if your DM allows you to attack with the shield.
Relevant text from PHB 195:
you can use a bonus action to attack with a different light melee weapon that you're holding in the other hand.
So, you get a total of 3 attacks.
Rage Bonus Does Apply
Rage is a modifier added to the damage, so the following rule (also PHB195 would apply):
You don't add your ability modifier to the damage of the bonus
attack, unless that modifier is negative.
Rage, however, is not an ability modifier, and still would apply. You wouldn't, however, add your strength modifier without the feat -- but since we're talking about with the feat your good on this front.
+3 to AC — Um... Maybe?
I'd argue it is either a shield or a weapon depending on how you are using it. I, personally, don't think a shield grants any more armor because it is being used a weapon. That said, using the shield is already against RAI, but might be allowed, and if the DM allows that, you could push for the +3 -- but I think it is a stretch.
Best Answer
According to the RAW in order to use a shield (6 lbs) and a dagger (light) weapon to attack in the same round, you would need the feat Dual Wielder.
The rules for two-weapon fighting (not the fighting style) on PHB page 195 stipulate that to get a second attack both weapons used must be light. In the weapon descriptions, the heaviest weapon that is classified as a light weapon weighs 3 lbs.
The Two-Weapon Fighting fighting style just allows you to add your stat bonus to your ability modifier to the damage from your second attack.
The Dual Wielder feat allows you to use weapons that are not light.
For clarification purposes, you must have the Dual Wielder feat to get both attacks when using the two-weapon fighting style if both weapons are not light melee weapons. (That is to say, if one or both of the weapons is not a light melee weapon then you absolutely need the Dual Wielder Feat to get the bonus attack.)
Further clarification, according to the section on two-weapon fighting (not the fighting style) in the PHB page 195 (emphasis mine):
In other words, to get the off-hand bonus attack both the weapons you are holding must be light melee weapons if you are only using the two-weapon fighting style without the Dual Wielder feat. The feat allows you to get the bonus attack even if the weapons are not light melee weapons. So in the above question, a dagger is a light melee weapon but the improvised shield is not, therefore the Duel Wielder feat is needed to attack with both weapons in the same round.
The addition of the Two-Weapon Fighting fighting style would allow you to add you stat bonus to the damage of the second attack.