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It was initially not possible, then later opened up, though a large collection of known surnames and personal names are reserved and cannot be registered as 2LDs. As far as I know the list of reserved names is unavailable, but you can run a whois query on any one of them to see it is reserved. E.g. check whois baxter.name and you get a response saying, "Not available for second level registration. Third level registrations may be available on this shared name."

I have the .name zone file and can confirm that no records for baxter.name appear therein, so it is not reserved only because a 3LD registration has ever taken place, it is reserved proactively just in case someone with that surname might want to register a 3LD under it.

Not at the very beginning, but then it became possible.

Yes it has been. I have had a second level .name for 20 years.

It was, I have one.