To answer the main question ("When was internet access first available to public in USA?"), one candidate would be FidoNet, an early popular BBS (Bulletin Board System).
According to this history, FidoNet had working UUCP gateways as early as 1986:
Although primitive experiments had been conducted earlier, in 1986
gateways between FidoNet and the uucp network, and hence the Internet,
became sufficiently reliable for production use.
UFGATE (UUCP-to-FidoNet Gateway) was the standard gateway for transferring mail and news.
Since FidoNet was an ad-hoc network of dial-up systems, with most nodes available to the public for free or at low cost, these UUCP gateways would have given users indirect access to the early Internet (assuming the UUCP host was connected to the Internet, which became more common as time went on).
This style of access, though, is not in the spirit of your second question ("where you open a connection to the internet, launch a web browser, and type in a domain name"). These gateways did not offer interactive or direct connections. FidoNet and UUCP were store-and-forward technologies, meaning the user's email might not be transferred for hours, or even days.
For more background, this page offers a good comparison of FidoNet and UUCP.