Alexandra Alvergne, Viri Lummaa 
Here, we review support for such conclusions and speculate on the
consequences of pill-induced choice of otherwise less-preferred
partners for relationship satisfaction, durability and, ultimately,
Central (supportive) argument in the paper for your claim
Hormonally based contraception inhibits the release of FSH and LH and
thus softens the normal mid-cycle peak in oestrogen level and elevates
progesterone levels (Figure Ib). These hormonal changes prevent
ovulation and mimic the hormonal state of pregnancy, which is
predicted to modify human female sexuality. In line with this, women
using the pill do not appear to exhibit typical oestrous sexuality,
but instead display preferences and interests similar to those of
women in the non-fertile phases (Table 1, main text).
There is emerging evidence that the use of the pill by women can
disrupt: (i) the variation in mate preferences across their menstrual
cycle; (ii) their attractiveness to men; and (iii) their ability to
compete with normally cycling women for access to mates. [...]. We thus
suggest the need for further studies using within-individual designs, and investigating whether general differences between pill users and non-users account for the effect of the pill on mate preferences. Whether the influence of pill use on mate preferences then interacts with actual mate choice is an open question
No final yes but a 'there is evidence.'
Anthony C. Littlea,Robert P. Burrissa,Marion Petrieb,Benedict C. Jonesc, S. Craig Robertsa 
They performed two studies. One using 55 women which were shown faces and had to rate them.
In our first study we experimentally examined change in preferences
following initiation of pill use. We recruited an experimental group
and a control group of women who completed two facial masculinity
preference tests with an interval of approximately three months.
One in which they analyzed 170 couples on the correlation between the masculinity of the male face and the usage of the pill when starting the relation.
we conducted a second study on an age-matched sample of 85 couples who
reported using, and 85 couples who reported not using, the pill at the
time of partnership formation. Standardized front-on neutral
photographs were taken of the men. We determined men's masculinity in
three ways: [...]
Our first study represents the first experimental demonstration that
pill initiation changes visual preferences for a trait associated with
mate-quality, complementing within-subject demonstrations that pill
use can change odour preferences for genetic similarity (Roberts et
al., 2008). Effects were only seen for preferences for opposite-sex
faces, suggestive that the effects of pill use influence mate
preferences but not general preferences for faces. [...] The second study builds on our experimental demonstration of changed preferences, documenting a downstream consequence of pill use during formation of actual partnerships, suggesting that altered preferences lead to altered mate choice. Original face images and computer generated images of women's partners, whom they met while using the pill, were judged as less masculine than those of women who met their partner when not using the pill.
My conclusion would be that there is strong evidence, and a good explanation, that women do change they preferences depending on whether they are on the pill or not. There is also a lot of related research on women preferences during the menstrual cycle which touches this topic as the pill influences said cycle.
Concerning the causation error you mention in part 2 I was unable to find a paper covering this topic but the first paper does say something on that topic:
Studies that compare divorce rates or marital satisfaction across
societies that differ by the prevalence of pill use but are similar
in, for example, social tolerance for divorce and use of
contraceptives might prove helpful in understanding whether marital
dissatisfaction owing to the effect of the pill on mate choice could
influence the duration and the stability of long-term relationships.
This indicates that - at least at 2010 - there was no such study to swing the conclusion one way or the other.