Good question, probably a good handful of ways to look at this as it sounds like a classic classification problem.
Define the label
First step, before you choose a model, would be to define the label. As you suggested, I would go for email opened = 1 and email not opened = 0. I would then think about throwing the email not sent (-1) data away, as I'm not sure you can learn anything about a probability of opening when it wasn't sent.
It would then be important to check the balance of the label: are they mostly opened or mostly ignored? This will influence any decision to add a class weight if it is imbalanced towards one class.
So time series could work, but I think another good angle might be to bin the send times into half hourly slots (or shorter, depending on your needs) and treat them as a categorical variable. You would then one-hot encode this feature. Anything else you can add to this will probably help, such as the kind of person it was sent to and what country they're in etc.
So it's classification, so have a play around with either logistic regression or something tree based (random forests, gradient boosted trees) to get something going quickly. It's usually a case of trial and error to find the best one.
You can then use traditional metrics of precision/recall/accuracy to measure how good it is. Make sure you know which metric is most important to your problem when optimising the model.