Well, those are tough questions to answer. Your thyroid is gone, so you’ll need to be on thyroid hormone replacement. That is a certainty. You won’t go hyperthyroid again unless you are taking too much thyroid hormone replacement, and it can’t possibly get out of control, because you’re being carefully monitored.
Will your eyes get better? Impossible to say. If you’re suffering from symptoms with your eyes that are related to the hyperthyroidism (spasms in the eyelids can retract them while you’re hyper, but that’s about it), then that will likely stop. The dryness and excessive tearing comes with having an autoimmune condition which, even without your thyroid, technically you still have (the antibodies exist in your body, but without their target). If you’re talking about TED, the two conditions (Graves’ and TED) are related to each other, but treating one does nothing for the other.
Will you get another autoimmune disease? Impossible to say. You are slightly more likely to end up with a second autoimmune disease, by virtue of the fact that you have one already, just looking at statistics. No way of knowing which it may be, or if you will get a second one at all.
Right now, focus on recovering from the surgery, and from the hyperthyroidism. Try not to get too caught up in "what next?" because it’s truly not possible to know everything that may come. Many patients benefit from stress-relief techniques, meaning yoga, talk therapy, reiki ~ whatever appeals to you, it’s worth a try.