They have their own reasons.
Surveys vary, but in the United states, as of 2017, about 35% of the population oppose gay rights.
No. In fact, they support gay rights, and always have.
Equal rights for gay people mainly affect gay people, but equality is good for any society. So in that sense, everyone benefits, even those that oppose equality.
The term "advocating homosexuality" is generally used by people who oppose gay rights. It is the mistaken belief that you can encourage people to be gay, which is impossible.
Words that describe people who oppose equality for gay people depend on the situation. Here are some options:scaredtraditionalconservativefundamentalisthomophobic
Each election, all of the Democratic candidates support gay rights and most (or all) of the Republican candidates oppose gay rights.
One certainly has the prerogative to think that way, but it is illegal in some countries to discriminate against any group of people because of their gender, race, background, creed, ethnicity or sexual orientation. Opposing gay rights would be synonymous with opposing human rights. This is not morally acceptable, but people definitely have the right to hate others.
"gay conspiracy" is a term used by those who oppose gay rights. It refers to the gay rights movement. These scared bigots believe that there is an organized plot to normalize the lives of some of their fellow citizens who have suffered for a very long time. The bigots do not want this suffering to end.
Yes, lesbian is the more usual name for a gay woman, but the word "gay" works as well. The term "homosexual" is outdated, and falling out of use, except among people who oppose gay rights.
Yes, all gay people in the United States have some rights, but depending on the state, they may be lacking some important rights, such as the right to marry, and the right to not get fired from a job for being gay.
Gay people marched. And voted. And picketed. And boycotted. And marched some more. And never ever gave up.
Gay people were not specifically considered upon the writing the Constitution, so there aren't any real "Gay rights" in the Bill of Rights. Although, the Bill of Rights still applies, I guess.