No, A never meant arm in 1A and 2A. The 1A and 2A designations were created by Alan Nagao for 1999 Worlds. At the time two handed play was seen as “better” than one handed so the divisions were labeled 1A and 2A similar to designations for minor league baseball or high school sports. It also mapped well to the number of yo-yos. When two yo-yo string tricks were created it was named 3A both since it was a combination of 1A and 2A and also kind of a joke because 3A would be “better” than 1A or 2A. When offstring and counterweight became contest divisions at Worlds in 2003 they were arbitrarily named 4A and 5A for no good reason. The A designations in general are confusing to the general public and really need to go away.
For the OP’s questions:
Basic string tricks have been around for a long time. The development of modern string tricks started in the late '90s with the rise of ball bearing yo-yos. The first time US Nationals awarded a National Title for one yo-yo play was 1997. Unresponsive play started in the early '00s. It started with natural response system wear on Renegades (and other yo-yos) making the yo-yo unresponsive which then made it easier to do some tricks so people started intentionally making their yo-yos unresponsive.
2A has been around for a long time.
3A traces back to Mark McBride’s velvet rolls in 1998. There were some basic 3A type tricks that are much older (ie, trapeze with one yo-yo and sleeper with the other) but I think most people would go with velvet rolls.
4A also dates back a long way in basic form as Duncan demonstrators could do a throw and catch. This is widely considered to be a ploy to get kids to try to mimic it which would result in a broken yo-yo and the kid needed to buy a new one. Modern 4A was made possible by the offstring bind/return which is credited to Jon Gates. I’m not sure on the year but it had to be before 1999 since Hironori Mii was doing full offstring freestyles by then.
5A was created by Steve Brown and first debuted to a wide audience at the 1999 Bay Area Classic.