I feel your pain man. While I haven't been in your exact situation I do know how frustrating it is getting told different things by different reps and just being given the overall runaround, as that is what happened to me with my whole upgrade debacle. I can tell you right off the bat that the first information you were given about payments on the AT&T NEXT plan was correct. You absolutely can make larger payments than required, as that is what I do regularly. You need to make your regular monthly payments, and for anything extra you wish to pay you need to call in to AT&T and just tell the rep you wish to make a payment towards the remaining balance on the phone. Tell them what you want to pay, give them your credit card, and you're all set.
As far as the unlocking is concerned, per the new laws they are legally obligated to unlock the device if you have fulfilled your contractual obligation and as a result, own the device. With the NEXT plan, the moment you pay the last penny owed, you own the phone and are no longer bound by contract. I would call back, and ask to speak to retentions immediately. If the rep who first answers tries to get you to explain your situation before they transfer you to retentions just state that you have explained it to too many reps already without resolution and are at your wit's end and just demand to speak to retentions, but be as diplomatic as possible. Once you have a retentions rep on the line, explain your situation in as much detail as possible, while also emphasizing the frustration and headaches this has caused you. And again, be diplomatic but let your frustration show through the tone of your voice. Mention the wasted time, and the runaround from all the people you have been directed to talk to. Tell them you are not happy with the way AT&T has been treating you, and that this is your last effort to work with them before you jump ship to another carrier. The retentions department has one purpose and one purpose only- to retain your business as a loyal customer of AT&T. They are the ones with the power to give larger credits (all my big scores have come from retentions) and they also have the power to escalate your case to whatever technical team is needed to solve the problem. Going this route is the absolute best chance of getting your situation resolved, if resolution is to be had. They'll get your phone unlocked (though it may take a few days, as it has in the past for me), and if you play your cards right, you may also get a decent credit to your account. If I was in a situation like yours, $50 would be my goal. If they offer you $50, take it and say thank you. If they offer you less, tell them you would feel better with a $50 credit. It's reasonable so they shouldn't have any problem with it. Give it a shot, at this point you have nothing to lose.
Good luck bud. Let me know what happens.