Cork and Cage vs Capped

750s tend to age better because there's less head space and oxygen that can get into the beer. As for caps vs corks, I'm not as practiced. I'd lay a cork on its side, like a wine bottle. Caps should stay upright.

 

Corked beers are more susceptible to oxidation because they don't seal as well as a capped bottle and cork taint is also a concern. The atmosphere inside of a beer bottle is fairly well regulated due to the liquid contents and the seal provided from the carbonation and whatever cork or cap is used so having the beer in contact with the cork really provides no benefit. In a fridge which is designed to keep things cold and dry you will have issues over a long period of time with the corks drying out. As far as laying the beer on its side to keep the cork conditioned and hydrated you have to remember that cork isn't very permeable so moisture from the contents of the bottle will not maintain the condition of the cork even if you lay it on the side.
As far as laying a beer on the side you expose a lot more of the beer to the gases inside the bottle so the beer will oxidize a bit faster.
In regards to which format ages better it's really up for debate as it depends on what qualities you are looking for. There is more liquid to disperse flavors like oxidation in larger bottles and due to the larger liquid quantity and similar gas capacity they are likely less susceptible to temperature shifts so they probably age a bit more gracefully and slowly than a smaller bottle but other factors will likely impact aging more than the size of the bottle and the way they top the bottle like the temperature you keep the beer at. Colder beer will age slower than warmer beer.