Let's address the issues one at a time.
1) What's the point in Local?
The goal is to give people the option to ignore the complexity involved with time zone rules. People are intuitively familiar with what these things mean. If you have a calendar on your wall above your desk: that's a LocalDate. If you have a clock on the wall: that's LocalTime. With respect to how they work in the presence of DST, they will automatically change time. For many use cases LocalDate and LocalTime is entirely suitable and people shouldn't be forced to put up with the unbearable complexity of timezones when they don't need to.
2) Why are they called Local?
As I mentioned above they represent the perspective of a clock on your wall or a calendar on your desk. They are local in this sense. I appreciate that these things might be a bit weird when you first get used to them and there was some discussion about renaming the classes during development. It was concluded that there were no better class names at the time. Furthermore if you looks at ISO 8601 [0] which is the most relevant standard to this API the same concept is also referred to as Local, Jodatime also uses the term Local.
3) Birthdays, Holidays and relative points in time
If you want to handle something like a birthday then there is a "MonthDay" class in JSR-310 which is designed for this exact purpose. It is a composite class of a month and a day and has an API consistent with the other classes in 310. If you want to handle 'relative' points in time then you might want to consider using the Duration or Period classes.
Hope that helps!