If it is used by a Jewish family, Foster is a Jewish surname. However, most so-called "Jewish surnames" are not exclusively Jewish.
No, it's Irish. There is a joke, however, about a Jewish or perhaps German immigrant who was asked for their name at Ellis Island. The immigrant, in Yiddish or German, answered "Ich habe vergossen" which means "I've forgotten," (really meaning, I didn't get the question). The clerk, without pause, writes down "Name: Ichabod Ferguson."
yes it is a Jewish surname
No, but a Jewish person can have any surname.
There is no surname that is used only by Jews. There are many surnames that are used both by Jews and non-Jews. Therefore, any surname that is used by a Jewish family is Jewish, and any surname that is used by a non-Jewish family is not Jewish. So any surname can be both Jewish and non-Jewish at the same time.
There are almost no surnames that are uniquely Jewish. If an surname is or was used by a Jewish family, then it is Jewish. If the same surname is or was used by a non-Jewish family, then it is not Jewish. And both can be true at the same time. For example, Cohen is an Irish Catholic surname with no connection to Judaism, yet Cohen is also a Jewish surname related to claims of descent from the Jewish priesthood.
no
Windsor
Not in its origin, but a Jewish person can have any surname.
The name Lewis is a surname for some Jewish families but it is also a surname for some non-Jewish families.
No.
No
There are almost no surnames that are uniquely Jewish. If an surname is or was used by a Jewish family, then it is Jewish. If the same surname is or was used by a non-Jewish family, then it is not Jewish. And both can be true at the same time. For example, Cohen is an Irish Catholic surname with no connection to Judaism, yet Cohen is also a Jewish surname related to claims of descent from the Jewish priesthood.
Although it is not a Jewish surname in its origin, any name can be a Jewish name if it is the name of a Jewish person