A VCU news release has the name of the (yet to be published) paper: “Genetics, the Rearing Environment, and the Intergenerational Transmission of Divorce: A Swedish National Adoption Study,”. The paper might be available through early access, dunno. Also, that web page says "study’s findings are notable because they diverge from the predominant narrative in divorce literature" (which probably answers some of your #2, but the see end of my answer).
This may be a bit of a stretch, but [the propensity for] infidelity is well studied, down to receptor and gene:
It is even possible experimentally to take a home-wrecking montane vole and make him behave like a family-oriented prairie vole. Using a virus as a delivery vehicle to transmit the vasopressin receptor gene, it’s easy to artificially boost the number of vasopressin receptors in the brain’s reward center, and make a male vole behave monogamously. The story for female voles is similar except that it is oxytocin, not vasopressin, that triggers monogamous behavior.
It doesn't work quite like that in humans, but
a study of nearly 7,400 Finnish twins and their siblings [...] found a significant association between five different variants of the vasopressin gene and infidelity in women only and no relationship between the oxytocin genes and sexual behavior for either sex.
On a related note, a dopamine receptor variant has been associated with one-night stands including infidelity manifested that way. So, if infidelity is a significant cause for divorce (rather than not getting along for other reason)... then it doesn't seem so far fetched divorce is influenced by genetics.
The issue I'm seeing with divorce and genes is that divorce is a rather heterogenous outcome of potentially many causes: infidelity, violence, substance abuse by one partner, infertility, and what not. Now it's possible that the main causes for divorce are all genetically determined to a substantive extent... but that's a rather roundabout way of saying something about human behavior.
It's actually rather difficult to even find good sources on the main reasons for divorce. One lists affairs/infidelity as the #2 reason, but another says that's far down the list. It might make a good question in itself here... A formal study (in the UK) also gives infidelity as #2 reason for relationships (formal or informal) breaking apart (#1 reason was "arguments").
I also found a 1996 twin study about heritability of divorce. Probably going for #1 reason for divorce ("arguments") they looked at MPQ of participants and found that
Positive Emotionality and Negative Emotionality factors were positively related to divorce risk, whereas Constraint was negatively related. In women and men, respectively, 30% and 42% of the heritability of divorce risk consisted of genetic factors affecting personality and divorce risk correlated largely as a result of these common genetic influences.
I also found a 2010 twin study of somewhat limited scope (all US male, relatively young). Frankly after looking at these last two studies I have the feeling the VCU news release presents their study as more anti-establishment/groundbreaking than it actually is. In particular, the last study I found says:
Consistent with past findings, results from the present study indicate that both getting married and ending a marriage by divorce are significantly influenced by genetic factors.
An earlier study by the same PI (Lyons) got picked up by BBC news in 2001. In contrast to their 2010 study, the old one did not find that marriage is genetically influenced (but divorce still was.)
I actually found a somewhat related twin study (also on a Swedish sample, like the VCU one) to have even bolder conclusions:
in this sample of older adult women, genetic influences on life events appear to be entirely mediated by personality.