@Savy: This is a version of the Coupon Collector's Problem.

Your work looks fine.

For n=10, the average EV per winning spin is SUM(1/n) {for n=1...10} = 2.93 spins.
The EV of total spins to hit each of 10 equally likely values at least once is 30 (10*2.93, rounded up).
The variance is 330 using the method in the link.

For the variance, I plugged the numbers n = 1 to 10 into Excel, created a column of p_i = 1/n, and created a column with (1 - p_i)/p_i^2. The sum of that column is 330, for a standard deviation of ~18.