A47
So much to unpack here. Plus this is really at least two separate and essentially unrelated questions.
All the Two-Weapon Fighting Style does is let you add your proficiency bonus to your bonus-action attack when using two-weapon fighting. It has no effect on what weapons can be used with two-weapon fighting. You may be thinking of the Dual Wielder feat.
To the first question, here's the text: "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."
Since a shield is not a weapon (let alone a melee weapon, let alone a light melee weapon), you cannot use two-weapon fighting if your "other hand" is holding a shield. I suppose... in theory... if you take the attack action to attack with a shield as an improvised weapon, then you could argue that at that instant, it's a weapon, so you get the bonus action to use with your sword. But as you note, a shield does not have the light property, so at a minimum the Dual Wielder feat would be required.
As for your Barbarian question, those features only require a melee weapon attack (or any weapon attack for Divine Fury). An attack with an improvised weapon is still a weapon attack, so there's no reason those features wouldn't apply.
Powers &8^]