Unarmed attack is an Improvised Melee Weapon. It does d4 + Strength modifier damage and does not have a proficiency bonus. See also What are some good stats to use when a player is attacking without a weapon (using fists)?

Initiative is d20 + initiative bonus.

I've forgotten the formal proof for this, but hopefully this is correct:

Consider a D6 (for the sake of concrete language).

When you roll a 1, you reroll the die and keep the result. This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 2, you reroll the die and keep the result (even if it's lower). This produces an average value of 3.5, and happens 1/6 of the time.

When you roll a 3, you keep the result. This produces an average value of 3, and happens 1/6 of the time.

And so on.

This gives the following formula for the average of the D6: \$ (3.5 + 3.5 + 3 + 4 + 5 + 6) / 6 = 4.1\bar{6}\$.

Working similar formulas for the other dice, we get this table:

\begin{array}{lccc} \hline \text{Die} & \text{(standard) Avg.} & \text{GWF Avg.} & \Delta \\ \hline \text{d4} & 2.5 & 3.00 & 0.50 \\ \text{d6} & 3.5 & 4.1\bar{6} & 0.6\bar{6} \\ \text{d8} & 4.5 & 5.25 & 0.75 \\ \text{d10} & 5.5 & 6.30 & 0.80 \\ \text{d12} & 6.5 & 7.3\bar{3} & 0.8\bar{3} \\ \hline \end{array}

Dice are independent. 2D6 will have an average value of \$2 \cdot 4.1\bar{6} = 8.3\bar{3}\$.

Common weapon average damage (Great Weapon Fighting):

\begin{array}{lcc} \hline \text{Weapon} & \text{Avg. GWF dmg} & \text{improvement w/ GWF}\\ \hline \text{Greatsword (2d6)} & 8.3\bar{3} & 1.3\bar{3} \\ \text{Greataxe (1d12)} & 7.3\bar{3} & 0.8\bar{3} \\ \text{Longsword (1d10)} & 6.30 & 0.80 \\ \text{Double-bladed Scimitar (2d4)} & 6 & 1 \\ \text{Smite (level 1, 2d8)} & 10.50 & 1.50 \\ \qquad \text{(+ weapon damage)} \\ \hline \end{array}

Observations:

• The ability works out to about a +1 to damage.

• It scales to almost a +3 when smiting. The more dice you add (high level smite, for example), the better the ability.^{See errata, below}

• The bonus is "swingy." It can range from a -2 to a +10 on 2D6, for example.

Errata

In April of 2016, Jeremy Crawford ruled that additional dice from abilities like smite can not be re-rolled by Great Weapon Fighting.

• Rules Answers: April 2016

• What exactly does Great Weapon Fighting allow you to reroll?