Is it your contention that a docket entry of "DISTRIBUTED for Conference of [date]" indicates that the case was simply assigned a conference date rather than purposefully selected by a justice for discussion at conference?
Pretty much, yes. It's been said that every case goes on the conference list, but that only cases on the discuss list actually get talked about and voted on. I think that makes perfect sense.
More importantly, before someone can say these cases have gone through anything but the most minimal procedure required to dismiss them, they have to show that something like the above is wrong. That somebody had to take some action based on the merits.
Yes.
Look at the Court's Order List from last Monday (or any Monday during the Court's term). There are literally dozens of cases in which certiorari was denied. Not all of them could have been discussed at the previous Friday's conference, but each of those cases, if you look up the Court's docket, will show a notation that they were distributed for that conference.
A case is "distributed for conference" under Supreme Court Rules 15 and 16 whenever the cert. petition has been responded to or when the time for response has passed without a response being filed. The individual Justices, or their clerks, review all those Petitions, and then circulate a "discussion list," which is not made public. Any Justice can put a case on the "discuss list"; if no one puts a case on that list, certiorari is automatically denied. (I am discussing the process for granting certiorari here; motions are handled differently.)
To be clear, I do not know if any of the Obama eligibility cases made it to the discuss list or not; we do know that none has had certiorari granted, that all of the motions for a stay or injunction have been denied, and that no Justice has recorded a dissent from the denial of cert. or the denials of the motions.