Not a Nazi anthem. The words were written in the 1840’s. The music was written in 1797 by the great composer Joseph Haydn and was used for the Aurtrian national anthem until the early twentieth century.
No, it predates the Nazi period. Also there's no good reason to suspect that singing the original first stanza at the Fed Cup was intentional rather than an unfortunate mistake (old versions are commonly found on the net and in books).
That stanza was eliminated from the anthem by the post-WWII government because of its recent association with Nazi aggression. The lyrics themselves were composed in 1841, though, when creating a united Germany from the various German principalities was a liberal cause [see Wikipedia]. There's nothing specifically Nazi about them. The anthem was adopted by the leftist Weimar republic in 1922 (YouTube video of Weimar anthem).
The outrage expressed by a player after the Fed Cup match was exaggerated. (Being in a bad mood after losing probably didn't help. :-) In the video of the event itself you can see at least one player smiling at the error rather than outraged (after 0:49 -- YouTube video of ceremony).