1. No, they are not independent. They are correlated: if I pick
a sleazy merchant, then they're both likely to burn out quickly;
if I pick a honest merchant, they're both likely to last a long time.
Formally,
Pr[X=1] = Pr[X=1 | sleazy] * Pr[sleazy] + Pr[X=1 | honest] * Pr[honest]
= 1/20 * 0.9 + 1/1000 * 0.1 = .1801
Pr[Y=1] = .1801
Pr[X=1 and Y=1] = Pr[X=1 and Y=1 | sleazy] * Pr[sleazy]
+ Pr[X=1 and Y=1 | honest] * Pr[honest]
= 1/400 * 0.9 + 1/1000000 * 0.1 ~= 0.00225.
Pr[X=1] * Pr[Y=1] = .1801 * .1801 ~= 0.03244.
We can see that Pr[X=1 and Y=1] != Pr[X=1] * Pr[Y=1], so X,Y
are not independent.
2. Yes, we are guaranteed this by linearity of expectation.
The lack of independence doesn't matter.
3. No, this is not valid. Variances only add if the random
variables are independent.