[RPG] What does losing a hand or arm to a crit mean if it was just a punch

Werewolves have all their immunities in every form.

Its statistics, other than its AC, are the same in each form. (MM p.211, "Werewolf," at "Shapechanger")

No*, attacks with the shield would not get +2 damage

The considerations/requirements:

No other weapons may be wielded in the off-hand.

This is straightforward: The fighter has some enemy by the collar with his/her offhand and is not wielding a weapon with it. So far so good.

The attack must be made with a 1-handed melee weapon

This is where things break down. As an Improvised Weapon, the shield is not classified as a Melee Weapon. At the moment of attack, the shield becomes an Improvised Weapon, which is legally separate from a Melee Weapon, even though the shield is used to make a Melee Weapon Attack.

Using the term "Melee Weapon Attack" does not change the shield's classification as an Improvised Weapon, it is used to distinguish the fact that the shield is not being used to make a Ranged Weapon Attack (with an Improvised Weapon).

Note that the Tavern Brawler feat does not reclassify the shield as a Melee Weapon and therefore is of no benefit for allowing the wielder to add +2 from the Dueling Fighting Style.

If the rules read "When you make a melee weapon attack," this would allow the fighter to add +2 damage from the fighting style to attacks with the shield. Unfortunately, the language requires that the fighter is "wielding a melee weapon in one hand" which s/he is not.

*One caveat: If an improvised weapon is "is similar to an actual weapon [it] can be treated as such." Therefore, if the DM allows the shield to be treated as similar to some Melee Weapon (an oddly-shaped club perhaps?), then the shield would benefit from the Fighting Style.