When moving a 5 gallon batch into secondary does it make a difference if it is into a 5 gallon carboy or a 6.5?
I say use any size it fits in.
In the secondary, there is unlikely to be any significant foaming unless you add a fermentable flavor such as a fruit juice. A five gallon carboy will serve fine.
Concerns about oxygenation in larger carboys, in my opinion, are largely unfounded. Small amounts of fermentation are still occurring in the secondary as well as natural outgassing from racking the beer out of the primary. Because CO2 is heavier than air, it forms a protective blanket over your beer, protecting it from oxidation.
The real danger of oxidation in the secondary is excessive splashing during transfer, not exposure while sitting still. But that is easy to minimize and doesn't have any bearing on the carboy size.
The current thinking among homebrew "experts" such as John Palmer and Jamil Zainasheff, as well as Wyeast and White Labs, is not to use secondary at all unless you're doing a true secondary fermentation by adding fruit or something like that. Just leave the beer in primary for 3-4 weeks, then package. That's the ultimate solution about what to use for a secondary!
I've never had a problem. I only use 6.5gal carboys, have for over 4 years.
I mentioned in a beer storage question that oxygen is one of the two beer spoilers. Minimize the exposure to oxygen by leaving little head space in your secondary vessel. If you keg, you can flood the secondary with CO2 before transferring. Carbon dioxide is heavier than air and will protect your beer from oxidation.
That being said, there's not much reason to chase a problem that doesn't exist. Oxidized beer has a wet cardboard or stale cracker flavor. If you don't taste those flavors, relax, don't worry and have a non-oxidized homebrew.
Doesn't matter much; but some will advocate having as little airspace as possible.
I use 5 gal carboys for secondary... when I use a secondary. Lately I have gotten lazy, except for lagers, I only use a primary.
I wouldn't put 1 gallon of finished beer in a 6 gallon carboy for sure, but 5 gallons in 6 or 6.5... probably not a big deal. And I never had a problem.
For the record I don't secondary that often anymore anyway.
I took a class at a local homebrew supply.. the person in charge of the class said the "beer" should almost fill the carboy, leaving little air in the carboy and limiting the chances of contamination. I followed this direction and the fermentation that took place started to enter bubbler..... so I poored a few oz out. Thanks to this episode I will not leave a good amount of room in the carboy for fermentation to take place.