A few caveats, before I get going. First of all, I do not presently practice in the Constitutional law/civil rights field. I am basing this primarily on my memory of my law school coursework on the subject, which is a few years stale, and what I've read in the news media and on sites such as SCOTUSblog. This isn't legal advice and if someone who does practice Constitutional law comes in here and tells me that I'm full of it, they're probably right :). Con law is my hobby, not my practice area.
To do more - and hopefully better-informed - theorizing, consider listening to the oral arguments, or reading a transcript. Although the Solicitor-General's argument is quite technical, the arguments by Mary Bonauto in favor of equal marriage and John Bursch in opposition are fairly accessible to a non-lawyer who is already familiar with the underlying issue.
Second of all, it's extremely risky to try to infer the ultimate outcome through the lens of the justices' questions. Sometimes a question may simply be that justice saying, "I want you to flesh out this issue more so that we can more easily deal with your opponent's arguments in our ruling." I've found that I get more traction from the tone of the questioning, but all prognostication is risky.
Continuing under the crumpled-up orange judicial robes...
Justices Kagan, Sotomayor, Ginsberg and if I recall correctly, Justice Breyer (I have trouble recognizing Breyer's and Kennedy's voices) made quite clear, when the Michigan AG was making his case, that they thought very little of his arguments. He was also pushed by iirc Kennedy and Roberts often, and, unlike Ms. Bonauto's arguments, which were often focused on what an old-school lawyer might call the majesty of the law, the overarching values of due process and equal protection, he got into the weeds of prior caselaw and rarely did it go well for him. It was, I believe, Justice Ginsberg, 82 years young, who took special exception to the claim that the government's interest in marriage focuses solely on procreation, noting that his argument essentially meant that 70-year-olds had no marriage rights - some surreal exchanges resulted there, but my hopes for a reference to The Fertile Octogenarian were dashed.
I had two takeaways on the arguments (and a couple gut feelings):
1) Justice Kennedy is either torn on the issue and/or he is acutely aware of the enormous weight on his shoulders, given that there are four justices who are 98% likely to support extending protections to equal marriage and he may well be the one to kick the ball over the goal line, if Roberts is a no vote (more on that below). He recognizes that this is not a case about traffic tickets or obscure federal regulations regarding migratory pigeons and that this will matter for a very, very, very long time, that it's likely that this case will be in every 1L con law book published for the next 50 years. But I also recognize that he's of a generation far removed from mine, where accepting LGBT rights has not been nearly as common or routine (when I found out my then long-distance girlfriend was transgender, my response was roughly, "Yeah, and?").
He seemed to be genuinely grappling with the idea that he would be making a change to marriage law in a nation that is often a legal bellwether for the rest of the developed world that as quoted, prior to the Netherlands in 2001, simply did not happen. Does that mean he's proud to be That Guy to make that change? Does he think that would live up to Earl Warren's legacy? Or does that mean that, in his view, he's reluctant to set off the one teeny tiny domino that will change the Western world (and possibly the whole world)?
Both are possible - I like to believe that he wants live up to Justice Warren's legacy in Brown. I don't always agree with Justice Kennedy but he undertakes his work seriously and with sobriety and an understanding of the effects of his rulings, unlike Justice Thomas, who IMHO treats this all as a fascinating parlor game, a 19th Century debating club (albeit in writing, since he never speaks at oral argument).
2) Is Justice Roberts looking for a way out, or a way in? His most interesting remarks were, to paraphrase, that maybe this whole issue could be answered without creating new case law by ruling that marriage bans are unlawful discrimination on the basis of gender, violative of the 14th Amendment, and calling it a day. That's a powerful tool that the SCOTUS has had all along but hasn't yet chosen to exercise - thus far, they haven't even directly answered whether or not LGBT is a protected class, let alone whether it's part of a gender-based protected class under the 14th Amendment, so nobody knows whether or not equal protection or due process will carry the day.
I haven't read the briefs, but based on the oral arguments, it's my understanding that the Obama administration supports equal marriage and weighed in primarily on the road taken to get there, supporting the equal protection route. There's two roads, equal protection (which is to say that everyone is entitled to equal protection under the law - similarly situated people must be treated similarly), and due process, sometimes called fundamental rights or liberties (a squishy area of the law where, for example, the right to abortion, contraceptives and, importantly, consensual sexual relations, i.e. gay sex, have been carved out).
I'm not a fan of due process arguments - I think that the absence of the word "privacy" in the Constitution has fueled many of the scream-fests over Roe and Casey, whereas "equal protection" is arguably the most important phrase in the 14th Amendment - there's no plausible way to argue that the 14th Amendment wasn't designed to add a new element, equality, into how the government interacted with its citizens. But that's a whole 'nother diary.
At times people have discussed the idea of essentially rolling LGBT protections into the protections people have against government discrimination on the basis of gender. The case law on that is fairly long as the modern Constitutional law world goes and also well-developed, and if the government ever does discriminate based on gender (for instance, a city giving female workers maternity leave but not male workers), it has to meet a high threshold, commonly called "intermediate scrutiny" or "heightened scrutiny." It is not like discrimination based on race, where almost (key word: almost) all discrimination will fail. But it's a standard with real, long, sharp teeth, and it would make things cleaner and more predictable than trying to carve out a whole new range of protections under due process/fundamental rights.
An equal protection finding would fuel follow-up cases, giving them an almost slam-dunk insurance that gay couples were treated equally in most relevant ways - tax code, veteran's benefits, adoption, foster care, any law or regulation where the government uses the word "husband," "wife," or "spouse." Roberts could write an opinion (or concur in the result) finally connecting gender discrimination to equal marriage and thus create an enforceable right without having to get at all creative like the Roe court arguably had to be.
Ms. Bonauto wants a ruling in her favor on both equal protection and due process grounds, and that's understandable. An equal protection ruling gets her heightened scrutiny and the instant protections of the myriad prior gender discrimination cases, but a due process ruling is likely to rocket her all the way up to strict scrutiny (on paper, when government action is subjected to strict scrutiny, it is almost always shot down by the courts - legislating in a field where strict scrutiny applies is the legal equivalent to tap-dancing in a minefield wearing steel-toed boots). But a due process ruling won't necessarily give LGBT people the immediate ancillary protections that an equal protection ruling is likely to bring.
Here's the problem - the SCOTUS doesn't like dicta, which is to say, language not necessary to answer the question that's been presented, and so it's rare (not unheard of, but rare) for the Court to rule on both questions. Thus far, the Court has elegantly tap-danced around the issues of heightened scrutiny/strict scrutiny; for instance lower courts have said that some anti-gay laws flunk even the weakest scrutiny level, commonly called rationality review, which is what it sounds like - the government must merely show that its action is simply barely logical and not driven solely by "animus" (what normal people would call "hatred"), and thus the Court didn't need to decide what level of scrutiny applied. If your protected class falls under rationality review, well, that sucks.
They're likely to rule on one or the other but not both. If I were forced to choose, I'd go with equal protection, which is what the White House picked. I'd rather inherit the history of all of the women's rights/gender discrimination case law and live with a slightly lower standard of review. With due process, she'd get strict scrutiny but have to write (and sue) the case law over from scratch on nearly every issue. In terms of ancillary rights (not just the physical marriage license), it'd be like the chaos that's taking place right now in Alabama, and it could easily take 10, 15, or even 20 years for there to be a stable body of case law on the subject. For comparison, see reproductive rights under due process, where even after 40+ years we're still trying to hit an ill-defined and moving target.
There is another possibility, that anti-LGBT discrimination be given its own category of protection under equal protection, separate from gender discrimination. But I'm going to go out on an admittedly shaky limb and say that such an outcome is unlikely. If they go for equal protection, odds are that Chief Justice Roberts will, based on his questioning, guide the case into the already-existing gender discrimination category. I can't imagine they'll take the California federal court's earlier emergency exit (the "it flunks all possible standards of review so we won't decide which standard applies" cop-out). This case squarely presents the question - despite decades of dodging the question, or perhaps because of decades dodging the question, it is almost certain that they will answer it directly this time.
So, there's the $64 trillion question (well, set of questions) - assuming that Justice Kennedy gives Obergefell a majority in favor of equal marriage, who will write that ruling? Will Justice Kennedy write it again, and will he tailor it narrowly or broadly? If Chief Justice Roberts is worried about being on the wrong side of history, will he merely concur in the result with his gender discrimination angle? Or will he take the proverbial bull by the horns and try to shape the opinion? Or will the world be turned upside down and it'll be the liberal four plus Roberts, with Kennedy in dissent or concurring in the result?