Let S be the event we're interested in -- all the multiple-of-6 lockers ending up open. I'm gradually going to repeatedly make observations of the form "Pr(these lockers all end up open) = 1/2 Pr(all but one of these end up open)", supported each time by the fact that some particular coin-flip affects what happens to exactly one of these lockers.
(By "coin C affects locker L" here, I mean that the result of flipping coin C changes what happens to locker L. Note that this isn't the same as saying that locker L gets (or might get) opened/closed after flipping coin C; for instance, no coin "affects" the locker with the same number, in my terminology, because after flipping coin n you then always toggle the state of locker n because n is both a factor and a multiple of itself.)
Coin 13 affects locker 78 but no other multiple-of-6 locker. Therefore, S happens iff (S ignoring 78) happens and coin 13 comes out the right way. So Pr(S) = 1/2 Pr(S ignoring 78).
Coin 11 affects locker 66 but no other multiple-of-6 locker. So Pr(S ignoring 78) = 1/2 Pr(S ignoring 66,78).
Coin 32 affects locker 96 but no other multiple-of-6 locker. So Pr(S ignoring 66,78) = 1/2 Pr(S ignoring 66,78,96).
Coin 45 affects locker 90 but no other multiple-of-6 locker. So Pr(S ignoring 66,78,96) = 1/2 Pr(S ignoring 66,78,90,96).
Coin 27 affects locker 54 but no other multiple-of-6 locker. So Pr(S ignoring 66,78,90,96) = 1/2 Pr(S ignoring 54,66,78,90,96).
Coin 20 affects locker 60 but no other multiple-of-6 locker. So Pr(S ignoring 54.66.78.90,96) = 1/2 Pr(S ignoring 54,60,66,78,90,96).
Coin 15 affects locker 30 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 54,60,66,78,90,96) = 1/2 Pr(S ignoring 30,54,60,66,78,90,96).
Coin 78 affects locker 6 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 30,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,30,54,60,66,78,90,96).
Coin 60 affects locker 12 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,30,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,30,54,60,66,78,90,96).
Coin 54 affects locker 18 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,30,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,18,30,54,60,66,78,90,96).
Coin 84 affects locker 42 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,30,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,18,30,42,54,60,66,78,90,96).
Coin 48 affects locker 24 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,30,42,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,18,24,30,42,54,60,66,78,90,96).
Coin 96 affects locker 48 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,24,30,42,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,18,24,30,42,48,54,60,66,78,90,96).
Coin 24 affects locker 72 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,24,30,42,48,54,60,66,78,90,96) = 1/2 Pr(S ignoring 6,12,18,24,30,42,48,54,60,66,72,78,90,96).
Coin 72 affects locker 36 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,24,30,42,48,54,60,66,72,78,90,96) = 1/2 Pr(S ignoring 6,12,18,24,30,36,42,48,54,60,66,72,78,90,96).
Coin 42 affects locker 84 but no other multiple-of-6 locker not being ignored yet. So Pr(S ignoring 6,12,18,24,30,36,42,48,54,60,66,72,78,90,96) = 1/2 Pr(S ignoring 6,12,18,24,30,36,42,48,54,60,66,72,78,84,90,96).
And that's all the lockers. We introduced one factor of 1/2 for each of the 16 lockers, so the probability is 1/65536.
(Highbrow version: what we're doing here is solving a linear system of 16 equations over the finite field with 2 elements, and the above is reducing the system to upper-triangular form.)