You can activate Action Surge, but it would waste it
The rules on surprise state:
A surprised creature can’t move or take an action or a reaction until its first first turn ends (remember that being unable to take an action also means you can’t take a bonus action).
Since you can't take an action, that means any actions (even if you have 2) and a bonus, since you can't take bonus actions when you can't take actions.
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):
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 you are holding in the other hand.
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.
Best Answer
Yes, you can use your Shield as an improvised weapon
The damage would be 1d4 plus your Strength bonus, the attack your Strength bonus, without your proficiency bonus.
No, you can not use it in addition
As it is not a light weapon, you can not attack with any other weapon in that turn unless you have the Dual Wielder feat.
What you could do
If you take Dual Wielder and Tavern Brawler, you can attack with a shield using a bonus action, and add your proficiency bonus to the attack roll.
What you should do
Take the Shield Master feat, get better DPR (advantage on attacks, plus Dueling Style) and better control, for half the price (spending only one feat).