David Blight gave the rough figure for radical abolitionists at the time of the civil war as 15% of the voting populace, mainly New England. The more middle-ground people would have been free soilers who just wanted to keep slavery from spreading.

I'm reluctant to say many of these people would have conceived of a slave or freed person as their equal. Class distinctions and beliefs in white supremacy were not considered contradictory to believing that slavery was a sin. What you find in most abolitionists is a sense of paternalism and desire to help those in need. That doesn't necessarily translate into seeing them as equals, quite the opposite actually but that gets into the idea of Toxic Charity and how paternalism warps people.

Source: David Blight's Race and Reunion, his free online Yale course, Bertram Wyatt-Brown's Southern Honor, reading a lot of articles and diaries from the time period, Toxic Charity

This is a very difficult question to answer and it depends on the time and place. Harriet Beecher Stowe, for instance, was extraordinarily patronizing at the time she wrote uncle Tom's cabin, came close to believing in equality on the eve of the civil war, and by the middle of reconstruction she did not believe black people were her equals. Other abolitionists never came close to believing in equality, while others tried to adopt a self-consciously Black perspective despite being white themselves. Examples of the latter are john brown and Gerit Smith. If you'd like to learn more about this, an excellent book on the topic is The Black Hearts of Men by John Stauffer

In Henretta's "America's History" it states that:

Almost half of free blacks in the United States in 1840 (some 170,000) and again in 1860 (250,000) lived in free states of the North. However, few of the enjoyed unfettered freedom. Most whites regarded African Americans as their social inferiors and confined them to low-paying jobs.

While Henretta doesn't specify exactly how many "most" is, it's safe to assume that he means the majority of white people.

Only a small number of African Americans owned land. "You do not see one out of a hundred ... that can make a comfortable living, own a cow, or a horse," a traveler in New Jersey noted. In most (northern) states, law or custom prohibited northern blacks from voting, attending public schools, or sitting next to whites in churches.

So, whether on a socially or legal basis, blacks were definitely nowhere near equal. However to say that there were absolutely no successful African Americans is a slight exageration.

Of the few African Americans able to make full use of their talents, several acieved great distinction. Mathematician and surveyor Benjamin Banneker (1731-1806) published an almanac and helped lay out the new capital in the District of Columbia; Joshua Johnston (1765-1832) won praise for his portraiture; and merchant Paul Cuffee (1759-1817) aquired a small fortune from his business enterprises.

These successes obviously weren't handed to them. In a political and social environment that was primarily against their whole equality, their achievements are truly surprising.

As black activist Martin Delaney remarked in 1852: "We are slaves in the midst of freedom."

So, based on that, I think that blacks being "inherently equal" may have been too liberal of an idea for the United States at the time. However, there definitely is no doubt in my mind that there were some who believed that blacks were equal to whites. However, to say so may have meant social ostracization, et cetera.

Henretta, James A., and Samuel T. MacSeveney. "Overlapping 
      Revolutions, The Free Black Population, Northern Blacks." 
      America's History. Boston: Bedford/St. Martin's, 2001.