Are supporters of gay rights considered apostates within the LDS Church? The official stance of The Church of Jesus Christ of Latter-Day Saints on homosexuality is:


  The experience of same-sex attraction is a complex reality for many
  people. The attraction itself is not a sin, but acting on it is. Even
  though individuals do not choose to have such attractions, they do
  choose how to respond to them. With love and understanding, the Church
  reaches out to all God’s children, including our gay and lesbian
  brothers and sisters.


The LDS church is in support of basic civil rights being extended to those who self identify as homosexuals. These rights include equal access to housing, medical care, employment and the like. They are not however in favour of homosexuals joining in marriage or gay cohabitation. This reflects their stance of, "People may not choose who they're attracted to, but they do choose how to act on those attractions."

Recently, because of the supreme court ruling which made gay marriage legal in the USA, in addition to the other countries in which gay marriage was previously legal, the LDS church found it necessary to update their definition of Apostasy to include same-sex cohabitation. Prior to the legalization of same-sex marriages, it was not necessary to be so specific, but in order to address the secular position that homosexual unions should be held in the same regard as traditional unions, the necessity has arisen.   

Amongst other questions, in order to receive a temple recommend, an individual must honestly be able to answer in the negative to the following question:


  Do you support, affiliate with, or agree with any group or individual
  whose teachings or practices are contrary to or oppose those accepted
  by the Church of Jesus Christ of Latter-day Saints?


Supporters of those gay rights which directly affect the definition of the institution of the family, specifically, those who support the institution of marriage between homosexuals, the cohabitation of homosexuals, or the adoption or raising of children by homosexual couples, are literally in support of practices that are contrary to the teachings of the church. They are literally in support of apostate teachings and practices.

Does this updated policy mean that all supporters within the church who believe that gay individuals have the right to live in gay relationships are guilty of apostasy?
I think the clearest thing I've seen said on this was a statement by Elder Christofferson of the Quorum of the Twelve Apostles (statement starts close to minute 9) in a press conference in January, 2015:

We have members, individual members in the Church with a variety of different opinions and beliefs and positions on these issues and other issues, reflect back on the Equal Rights Amendment years ago, this isn’t the first issue, in our view it doesn’t become a problem unless someone is out attacking the church and its leaders, if that’s a deliberate and persistent effort, trying to get others to follow them, to draw others away, trying to pull people out of the church, or away from its teachings and doctrines. That’s very different for us, than someone who feels one way or another on a political stance or a particular action to support a group, Affirmation or any others that you named, these are things that there are fine lines here and there.

This is consistent with one of the other actions specified as apostasy in the handbook, even before the recent changes (see the unhighlighted portions of the handbook at the bottom of this article):

As used here, apostasy refers to members who:

Repeatedly act in clear, open, and deliberate public opposition to the Church or its leaders.


So the takeaway from this would be that, unless you are persistently and publicly voicing your opposition to a teaching of the church, you are not necessarily considered to be in apostasy.
One final note, in relation to your quotation of the temple interview questions: Another contributor stated, in response to a different question about supporting polygamy and standing within the church, that &quot;temple worthiness is on the far opposite end of the spectrum from excommunication: one who may not enter the temple may not be near disciplinary action.&quot; Although about excommunication, and not apostasy, I think the same idea applies here.