Are there any accounts of of Indians being forced to use certain facilities or were they free to use any facilities they wished?
Foreign dignitaries were often treated well. A Sri Lankan man named Chandra Dharma Sena Gooneratne wrote some anecdote of his experience in Saturday Evening Post: "Any Asiatic can evade the whole issue of color in America by winding a few yards of linen around his head...A turban makes anyone an Indian." He also wrote once he sat in "colored" coach of a train and the conductor insisted that he should sit in the white section. The conductor gave up after Gooneratne questioned how the seats were different.
Later a black Lutheran minister named Rev. Jesse Routté visited his family in Alabama in the 40's, he said "I was Jim Crowed here, Jim Crowed there, Jim Crowed all over the place. And I didn't like being Jim Crowed." So the next time he visited Alabama, he found a turban and robe and spoke in Swedish accent he learned from Swedish Lutheran theology school and everyone treated him like royalty thinking he is a foreign visitor. Here's a newspaper clipping with a photograph, and here's a New York Times clipping. At a fancy restaurant he asked the staff what would happen if a "Negro gentleman comes in here and sits down to eat." The reply: "No negro would dare to come in here to eat." "I just stroked my chin and ordered my dessert," he said.
This isn't a direct answer to your question, but it is in the same vein.
You might be interested in the case of Chang and Eng, two conjoined twins of Chinese descent who immigrated to the United States in 1829 (it is from them that we get the term "Siamese Twins"). They toured the United States from 1829-1839 and then settled in South Carolina where they raised a family had 21 (!) children between them.
However, what is interesting is not specifically this, but that they claimed "normative whiteness" (A term I am stealing from Joseph Andrew Orser's book and dissertation about the twins). They claimed for many years to be white, and owned property and slaves in the south. This worked for a time, but after their death in 1874 their family was again subjugated due to widespread anti-Chinese sentiment.
This is just an illustration, but in general the American Legal system, after adopting the "one drop rule" in many states, had trouble classifying individuals. This is why a number of individuals of "mixed" race took to claiming that they had Native American ancestry, in order to escape this rule, with varying degrees of success. But in general, people who were not white were considered black.
I cannot remember any specific examples regarding individuals from India. Unfortunately I just lent out the book that I had regarding this. It is After Identity by Georgia Warnke, she is a philosopher/political theorist, but presents a compelling history of the way the legal system treated "troublesome" individuals who did not easily fit into the two major social binaries, white/black and male/female as a way to argue against the objective existence of racial and gender categories.
I can only speak to the legal rather than cultural aspect of your question. The Supreme Court of the United States was called upon to determine the whiteness and non-whiteness of various races as a matter of law during segregation. In United States, v. Thind, 261 US 204, (1923), the Supreme Court specifically considered whether "a high caste Hindu of full Indian blood [was] a white person within the meaning of [the Naturalization Act limiting naturalization to whites and Black Africans]." Id.
Pursuant to Section 2169 of the Naturalization Act of 1906, the Act applied "to aliens, being free white persons, and to aliens of African nativity and to persons of African descent. The Supreme Court rejected Mr. Thind's argument that he was Caucasian based upon the "Aryan theory," which you might recognize if you are familiar with Hitler's version of history.
The Supreme Court observed the term "Aryan" had to do with linguistics, rather than with physical characteristics. Indeed, the Court observed that an observer in America would find several African Americans speaking English who, nevertheless, were not white.
The Court did acknowledge the possibility of an intermediate classification, but no case or statute created a "tiered" racial hierarchy like what existed in South Africa or, for that matter, India (As Thind claimed).
The Court observed:
"It may be, therefore, that a given group cannot be properly assigned to any of the enumerated grand racial divisions. The type may have been so changed by intermixture of blood as to justify an intermediate classification. Something very like this has actually taken place in India. Thus, in Hindustan and Berar there was such an intermixture of the "Aryan" invader with the dark-skinned Dravidian.
In the Punjab and Rajputana, while the invaders seem to have met with more success in the effort to preserve their racial purity, intermarriages did occur producing an intermingling of the two and destroying to a greater or less degree the purity of the "Aryan" blood. The rules of caste, while calculated to prevent this intermixture, seem not to have been entirely successful." Id.
While racism is abhorrent in all its forms, one may appreciate a certain irony here. The "high caste" Hindu's claim of pure Aryan descent, based on a system of segregation stretching back to antiquity, was rebuffed by Sutherland's casual observation of Respondent's skin tone.
Justice Sutherland ended his opinion by finding that Asian Indians were not white. You may find the full public domain source on google scholar under case law.
You may also be interested at looking at other cases on the subject. The final score was that Hispanics, Jews, and Middle Easterners were legally white and able to naturalize.
A Japanese man in Ozawa v. United States, 260 US 178, (1922), failed to be declared white based on a literal argument as to the color of his skin being lighter than many Europeans.
Middle Easterners were generally considered white. See United States v. Cartozian, 6 F.2d - (Ore. 1925). While none of the Supreme Court decisions come right out and say it, I think a fair assumption is that the Judges had a hard time assuming the characters in the Bible would have been barred naturalization based on U.S. racial law.