Aloha Tatiana! I wish your questions had straightforward easy answers – and if you are government funded it kind of is. (See here: https://www.ada.gov/websites2_prnt.pdf) if you are not government funded, this is something that is currently being debated in the legal system. That being said, the WCAG 2.0 guidelines are a good set of guidelines to help protect yourself if you feel you should. (You can see more on WCAG here: https://www.yokoco.com/find-out-how-to-make-your-website-compliant/)
There are exceptions to this title; many private clubs and religious organizations may not be bound by Title III. With regard to historic properties (those properties that are listed or that are eligible for listing in the National Register of Historic Places, or properties designated as historic under state or local law), those facilities must still comply with the provisions of Title III of the ADA to the "maximum extent feasible" but if following the usual standards would "threaten to destroy the historic significance of a feature of the building" then alternative standards may be used.
Most recently, however, pizza chain Domino's has been brought under suit for their website not being accessible for specialty ordering. The U.S. Supreme Court refused to review the case, instead upholding the decision of the 9th U.S. Circuit Court of Appeals who said the “alleged inaccessibility of Domino’s website and app impedes access to the goods and services of its physical pizza franchises—which are places of public accommodation.”