How are elasticity of demand, marginal revenue, and total revenue connected?
Marginal cost is irrelevant to total revenue. It's relevant to profit, but not total revenue.

Total revenue is just price x quantity.

Marginal revenue is a monotonic decreasing function. That is, as quantity increases, marginal revenue decreases. So, let's take the quantity Q to be the point at which marginal revenue is zero.

For quantities below Q, marginal revenue is positive, so total revenue increases with quantity.

For quantities above Q, marginal revenue is negative, so total revenue decreases with quantity.

Hence Q is the turning point of the curve: it is the point of maximum total revenue.

Elasticity of 1 means that a tiny % change, $\Delta$ in price in one direction (equivalent to multiplying price by $1+\Delta$),  will be met by a reciprocal proportional change in quantity in the other direction, equivalent to multipying it by $\frac{1}{1+\Delta}$: that's straight from the definition of elasticity. 

So total revenue, the product of price and quantity, remains the same:

$$(1+\Delta) P \times \frac{1}{(1+\Delta)} Q = P \times Q$$

Hence marginal revenue always 0, when elasticity is exactly equal to 1.