What is the Republican Party's stance on entrenched inequality; gender, racial or otherwise?

Your assumption in your question is wrong. Inequality  in the work force is not a problem but a normal state. Since no two people are totally equal in abilities, their value on the work force would also be unequal because employers pay according to the productivity of their workers. You allude to the fact that there are other factors an employee faces such as cronyism and opportunities by relationships. Those exist because those employers  have the power to act in that way.  There is no mechanism within a government with limited power, that  can change that. That would be  a limitation on freedom to choose. Affirmative action started as a campaign to get private companies to hire minority groups. It was a voluntary campaign and it relied on public relations as a motivation. Somehow it morphed into a government policy which was never  the intention of affirmative action when it began.

I can tell you what it shouldn't be. Quotas.  Quotas make everyone unhappy. Quotas flooded the sub prime market with worthless paper in 2008. I can tell you what it should be.  A nation where people are judged by the content of their character and not the color of their skin.

1. To the extent that it exists it should be eliminated. 2. To the extent that private industry can eliminate it, it should. 3. To the extent that private industry can not or will not eliminate it, state and/or local government should step in. 4. Such involvement should be narrowly tailored (for instance, it should avoid interfering in businesses with less than, say, 10 employees or religious groups who refuse to hire from outside their religion; in addition, local chapters of, say, the National Organization of Women shouldn't be forced to hire a 50% male workforce). 5. Regulations implemented by state and/or local governments should be regularly reviewed to determine their cost/effectiveness/need.

[Because the question has changed, I've revised my answer to better fit it.] Mankind soon learn to make interested uses of every right and power which they possess, or may assume.  The public money and public liberty...will soon be discovered to be sources of wealth and dominion to those who hold them; distinguished, too, by this tempting circumstance, that they are the instrument, as well as the object of acquisition.  With money we will get men, said Caesar, and with men we will get money.  Nor should our assembly be deluded by the integrity of their own purposes, and conclude that these unlimited powers will never be abused, because themselves are not disposed to abuse them.  They should look forward to a time, and that not a distant one, when a corruption in this, as in the country from which we derive our origin, will have seized the heads of government, and be spread by them through the body of the people; when they will purchase the voices of the people, and make them pay the price. --Thomas Jefferson, Notes on Virginia, 1784 How much mockery did those of us on the right receive in the lead-up to the 2012 election over "our desire" to kill Big Bird? The mockery masks gross ignorance. Just as many a progressive friend quoted to me that our big yellow friend costs mere pennies per person, that is always the nature of public goods and the reason they are so open to abuse: "Why, if we did that, it would only cost 17 cents per citizen... only $1.12 per voter... only $17 per taxpayer... Everything becomes feasible, and the final analysis of ensuing disaster becomes akin to "which raindrop caused the flood?" My same progressive friends cast the Republican Party as "the Party of NO!" I wish. They strike me as "the Party of NO! Well... okay then." At the same time, the Party of Jefferson has stood poor Tom on his head. They have taken the http://en.wikipedia.org/wiki/Rent-seeking he was railing against above to artistic heights. The US has always had (and I hope always will) ad hoc grievance groups. Now we have a priori ones, and divvied up along the most mundane lines: I am a [black, brown, red, yellow, white] [female, male, both, neither] who likes [vulvas, vaginas, rectums, penises, everything, nothing] [*If you have selected white, male, vaginas, do you have an appropriate sense of guilt? yes, no]. And they love the Democratic Party! Because that party caters to them. As a cool libertarian rather than a stiff Republican, I often am in social settings where the assumption is everyone is progressive and cool and the guard is down. People are just being people. Something will be said or an image of Dick Cheney will flash on TV, and suddenly everyone is in grievance-group mode, all puffed-up and spouting pre-scripted rationalizations about their "deservingness." Comic but sad. How about instead we just be people? We prize all, in our full glorious range of diversity and uniqueness, exalting everyone, but doing so in sound, healthy, productive, affordable ways?

I’m not qualified or authorized to speak for the Republican Party but I, myself, don’t believe that inequality based on racism or gender is a “huge” problem.  Obviously the term “huge” is subjective.  And I’m not saying that racism doesn’t exist in America.  The fact that in 2008 97% of the black-electorate voted for the demonstrably lesser-qualified black candidate very strongly suggests that racism is alive & well.  I believe that your statement that Republicans are against (equal) “opportunity” to be patently false.  The fact that the Left wants to move lesser-qualified individuals to the head of the line does not mean that they favor (equal) opportunity more than the Right does.   In both my military & civilian careers, spanning thirty-eight years, I can honestly say that I’ve never seen a lesser-qualified white person promoted over a minority.  In both careers I’ve had plenty of supervisors who were black, Hispanic and/or female.  I’ve read that a larger number & percentage of women are graduating from college and that during this last recession men were hit harder than women.  As for the gender-gap in pay, I don’t believe that it exists:   http://www.youargue.com/index.php/arguments/browse/40-the-myth-of-gender-wage-discrimination.html   http://www.cbsnews.com/news/the-gender-pay-gap-is-a-complete-myth/

The Republican party is not "for" inequality.  The Republican party is "for" freedom.  Sometimes freedom results in inequality, and it's the left in our country that thereby concludes that Republicans are "for" inequality. This is not a hard concept to grasp.  Republicans want to guarantee equality of opportunity.  That's in the US Constitution. Democrats want to guarantee equality of outcome.  That requires a huge loss of freedom to ensure.  And if equality of outcome is ever achieved, it will mean a very low level of equality indeed.  Just look at how productive communism was in both China and the Soviet Union.  I note that both of those nations have abandoned Communism, by the way.

Bigots do not find a humanistic party to their liking. They therefore join a party that pays less attention to humanistic value, and more to the freedom to think, and do, what they want. The Republican party is basically for a republican form of government, which we are supposed to be; the Democratic party is for people and our relationship with each other, therefor they think the USA is a democratic nation, which it aint. I think that Intellectuals, and idealists, are likely to be Democrats, and people who deal with reality, not inclined toward humanistic idealism, are Republicans. So Demos want equality and Republicans want what they can earn, and don't like "takers." Demos are inclined toward guarantees in life, Repubs don't believe guarantees are fiscally possible, and destructive to the spirit. I am in the middle; balanced!