By waiting until the home market closes for the day, your trade incurs less risk for the MM
I would think it incurs more risk - namely if some news occurs prior to the home market open. If you trade while the home market is open, the MM can trade simultaneously if they so desire.
Using Fidelity I've traded on the pink sheets a couple of times and I didn't get a particularly good price either time. Fidelity now allows trading on a few foreign exchanges directly. Interactive Brokers, as discussed previously, allows trading on many more foreign exchanges. Their trading platform is fairly complex though, so takes a bit of getting used to.