I can't give a full comprehensive answer because I don't really know where to look without just manually looking through every season for every team ever
and I'm not quite that crazy
But I can answer the first question.
Yes, NBA teams have started 6-22 or worse and still made the playoffs in that year.
The first example that comes to mind is the
1952-53 Baltimore Bullets, who made the playoffs despite starting 6-22 and finishing with a record of 16-54. They remain the worst record of all time to actually make it into the playoffs.
Another team that did it though, which is the only other one I can remember, is the
1984-85 Cleveland Cavaliers, who
also started the year exactly 6-22, but finished with a 36-46 record and made it into the playoffs, where they lost 3-1 to the Boston Celtics.
Neither of those teams made it over .500 despite that start, but there could be others that I just don't know about/couldn't find with an easy Google.