Antero will not tell you who all the owners of the mineral rights are. Their usual line is that they paid a lot for that research and they’re not going to give it away for free. Additionally, they have to be careful about privacy. When I was a landman, I found family members that had fallen out of touch. If one requested to be put back in touch I would make sure to contact the other one first and discuss it with them. If both agreed to get back in touch I would give them each other’s contact info. If not, I wouldn’t. You never know who might be a crazy wack job.
If you really do want to find all the other heirs it’s going to take a lot of work. You’ll need to get on Ancestry.com (most of the landmen use this) and start researching the descendants of your great-great-grandfather. Sometimes this is easy, but not often. You’ll learn a lot, though, and genealogy can get pretty addictive. Once you’ve identified heirs you can look their names up on the county records and see if they’ve signed leases or sold their rights.
The taxes probably will increase, but they may not increase as much as you think. What will be a surprise will be the taxes on production, which will come for the first time two years after production starts. Antero will report to the county, and the county will assess you based off the reported royalties. It’s small, about 2.5%, but it will come as a surprise. As far as working with the other heirs, that won’t be as hard as finding them. Once you’ve found them I’m sure you can work something out. At very least you can request that the county split your interest out from their interests so you get taxed separately. The county will usually require you to get a lawyer to review the title and submit a letter to the county, but you’ll have done most of that work already so it won’t be awfully expensive.