Hi Susan,
You are doing the right thing by investigating options before making a decision. There is a lot of horizontal drilling going on in Doddridge and Antero is trying to get your mineral rights in order to secure up large tracts of land into units where they can put in another well. They (and probably all the oil and gas companies) approach family members like you who live out of the area, inherited rights from an old lease, and probably didn't even know you had rights, with the hope that you will give in to THEIR best case scenario and sell them for what they consider, small change.
I am not an expert, but I have been reading a lot in this forum and I can give you my opinion. I recommend you form your own opinion, but I think it would be most wise to keep the mineral rights. Antero is telling the truth that you will have to help pay to process the minerals, but what they are not telling you is that you will also profit after paying. From what I understand, you do not have to come up with the money to help process the minerals, it is deducted from what they pay you in profit. That is why they want your rights because there is a potentially BIG profit involved.
If you decide to keep your rights and sign a lease with Antero, it is most advised by everyone I know on here who has gone through this lease signing ordeal to either hire a WV oil and gas lawyer to represent your interests on the lease or to research and learn everything you can about leases and terms before negotiating with them and signing. Researching yourself is very time consuming and stressful (I know from experience).
The oil and gas companies are famous for using tactics to pressure you into signing away or signing a lease quickly and for much lower than you can get if you negotiate terms. The leases and deals they offer you do not benefit you in ANY way, but are just enough to tempt you to to sign. They may tell you there is a time limit, but there really isn't. They may tell you this is their best offer, but without a lawyer behind you, you have no way to know that - and it probably isn't. They will tell you half truths, like what they said about how you will have to share in the costs, but they will not tell you the whole truth because you would not want to sell to them or if you still did want to sell, you would ask for a lot more money.
Bottom line is, your mineral rights are very valuable even if they seem like a small interest (1/16th). Do not be hasty in giving them away.
As for back royalties, there is a website to look up unclaimed property held by the state. If the old well was giving out royalties and the lease was reassigned to someone else, but the leasing company never got a change of address or affidavit stating the new ownership, then they would have sent the royalty payments to the state to hold. You can go there and type in some of the names you have and see what comes up (make sure to scroll down after hitting search because it looks like it didn't do anything but the list is below and you can page through it). Here is the website: http://www.wvsto.com/dept/UP/Pages/ClaimInformation.aspx
Also there are a few places to research specific wells and their productivity. Here is one: https://apps.dep.wv.gov/oog/wellsearch_new.cfm You don't need to fill in all the fields, just one.
Here is an interactive well map: http://tagis.dep.wv.gov/fogm/ that will give you info like the types of wells, locations, and production history.
Keep asking questions and there are lots of people here who are happy to help.
Stephanie