Although I have been to Beijing a decent few times I am not going to claim to be an expert.
Beijing is a large spread out city with Hilton properties spread from East to West. I would say with your requirements although good value you can rule out the Double tree.
For access to the airport and public transport for the Great Wall you can't go wrong with The Hilton Beijing. Closest Hilton to the airport barring the airport Hilton, 10 minutes to Dongzhimen station which has the bus to the Great Wall and almost beside a metro station which I think is two stops from Forbidden city/Tiananmen square. It is usually reasonable value has a good club lounge and rooms are decent if not slightly dated. This is were I have been staying on most recent trips to Beijing along with Double tree.
I only have limited experience with the Conrad and while a great hotel it would probably be wasted on what sounds like Is going to be a whistle stop tour of the city. Have not stayed at the Hilton Wangfujing but it gets good write ups.
I cannot help regarding gold treatment at the various properties but I know the lounge at The Hilton explicitly has a sign saying Diamond and exec floor guest only. Don't know if that means they do not upgrade golds. For such a short stay though I would not limit yourself to a Hilton. If you are wanting to take public transport everywhere the Holiday Inn express Dongzhimen station might be an option. Can arrive, drops bags, jump on bus to Great Wall and back again and then Dongzhimen is well connected for the tourist sites in th city and also the airport express.
Check out all the relevant property threads on FT as that should give more info regarding gold treatment etc. Hope this helped a little but I know more people will have better information. Enjoy the Great Wall. It really is incredible. I reccomend not going to Badaling and if you are wanting to use public transport go to Mutianyu as it is fairly well connected. Have a great trip!