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.