As it with many things, the answer to your first question is basically, "it depends." In this case, it depends on where the immigrant is from, who they know when they arrive, and where they're heading. That is, if they were from a country with a history of emigration to America, which meant time and people to establish structures for supporting new arrivals, then yes, there were ESL courses.

From a previous response on a similar topic:

Immigrants typically decided for themselves how, when, and to what degree they became "American." And it was very time, family, and location-specific. Some parents made the choice to never speak their native language in front of their children to as not impede their Americanization while other parents kept their child home if the school insisted children only speak English. Girls from one tenement building might see their brothers go to school while they were required to work in a factory. Boys in the next district were expected to work alongside their parents and didn't attend school until the local district hired a truancy officer explicitly to track them down. However, as states and school districts moved to a per pupil funding structure, more and more children were moved from the factory to the classroom, even, at times, overriding parents' wishes.

In some cases, most notably for German families, there were enclaves across the east coast and Midwest where they would be surrounded by the German language, food, and cultural touchstones. Their children could go to a public school (mostly for free, funded by tax dollars, or a small fee which went towards the teacher(s) salary and school upkeep) and learn English and the lessons of Americana. In many cases, teachers taught stories about Columbus, Washington, and the founding of America in both German and English (more on why schools teach "bad" history.) There are documented instances of teachers chiding children for speaking German but there were also cases where teachers allowed children to speak whatever language they wanted, as long as they were kind to children who spoke different languages. (Getting along with people from different backgrounds was/is seen as an American trait.) In a real sense, this meant that the immigrants were American from the moment they arrived on American soil.

For some Scandinavian immigrants, the presence of a school with a well-stocked library and trained teacher was the prime motivation for continuing out of New York City and farther west. In many cases, immigrants were joining family members who had built those schools and recruited teachers from the east coast to staff them. Italian and Jewish immigrants faced different sets of choices based on how the community they arrived in viewed their "otherness" and their proximity to whiteness (more on that in a bit.)

So, in effect, if a newly arrived German immigrant settled in a German enclave with an established social network, they would likely find themselves learning English with the help of others, including formal courses in learning English. Turning the lens around, American religious organizations, schools, and social networks routinely offered English courses to newly arrived immigrants. As an example, Hull House in Chicago offered multiple levels, including classes children and adults.

All of that said, social norms around bilingualism and multilingualism varied from place to place. In some communities, immigrant children could attend a tax-player funded public school in the early 1900s and not hear a single word of English as every teacher and student spoke the same language as them. There were other schools where children spoke dozens of different languages and teachers taught in English with little or no concern if a given student understood what had been said. Finally, it's worth stating explicitly that an immigrant's English abilities were used to justify xenophobia and racism regardless of what direction it went. Newspaper editorials from the West Coast warned against "overly" educated Asian immigrants while editorials from the American South rattled on about non-English speaking Mexican immigrants.