1. She has a Jewish sounding surname.
Not especially.
In the US, it is. Many "Jewish" surnames are either German or Eastern European in the US.
On the other side, many surnames "can be Jewish". BROWER is an example, and I have no Jewish ancestry before 1800; the Brower ancestor that came to the New World in the 1640s was a member of the Dutch Reformed Church.
The first Jews of New Amsterdam almost all had Spanish or Portuguese surnames.
Well, there were Jews in the colonial period, the Solis-Cohen family in Phila, for example, but
most of the Jewish population of the US have ancestors that came here in the 1800's, generally from the Germanies or Eastern Europe.
As pointed out, I know people with the surname Holtzman, Abrams, Bergman, Jacobs, Rosensteel, and Rosenberger, none of whom are Jewish, nor have Jewish ancestors (and my old girlfriend, who was Jewish, had the surname Brown).
Many people assume, from the name, that they are Jewish.
I can see people making that assumption about Bachman. Coupled with Bachman's fairly pro-Israel stance, and that she worked on a kibbutz in Israel one summer, I can sort of understand why someone might jump to that conclusion.