Hi... I'm sorry you've gotten the diagnosis of hyperthyroidism. Are they sure it's Graves Disease? Did they test the TSI antibodies to be sure?
Liver damage is rare side effect of methimazole. Nobody can guarantee that it's 100% safe. Treatment is, often, for 12-18 months, typically, not for life and the longer you take the medication, the more likely to have adverse effects.
One of the symptoms of hyperthyroidism is a condition called Thyroid Eye Disease (TED)... From what I know, it's more prevalent in those not treated with RAI than in those that have treated with RAI. Again, though, no one can guarantee that you can't/won't get it.
Surgery is an option that some people opt for over RAI... both of them will leave you hypothyroid and dependent on replacement thyroid hormones for life, but there are worse things. There are several of us on the forum who are on lifelong hormone replacement due to Hashimoto's, which also destroys the thyroid.
Graves Disease is an autoimmune condition and once we have an autoimmune condition, we're considered to have it for life, however, once the thyroid is gone or disabled the antibodies will go into remission and will no longer cause problems.