1. Is this part of my ancestry Slovak, or Jewish (Israeli)?
Slovakia and Israel are polities. Since you can't be from
"Jewish", and Israel is a very young state (probably younger than your grandfather), I would say you're an American of Slovak heritage.
2. Are people who are mixed European/Jewish (white) considered bi-racial?
No. Or at least I don't think so outside of certain Zionist circles who want to push "Semitic" origins of people who were mostly converts of Central and Eastern European origins for political purposes. I mean, since you obviously have everyone from Yemeni and Iraqi Jews (i.e., actual Middle Eastern/Semitic people), to the Beta Israel/Falasha of Ethiopia, to Iberian (Spanish/Portuguese, "Sephardic") people, to even whiter folks claiming "Jewish Ethnicity", it seems completely elective and more of a political statement than anything if someone decides to identify as ethnically
So I guess a more precise answer could be: No, of course not...but also yes, depending. But it's important to realize this "bi-racial" identity isn't quite like others...if you're half black and half white, or half Latino and half whatever, you probably deal with issues of being able to "pass" for one or the other, whereas people who choose "Jewish and something else" are making an elective choice to claim an identity that has no sure phenotypical or cultural/linguistic markers that might cause them difficulty. I was friends with a Jewish guy of Polish and Ukrainian origins in college and after he told me that he's not really "into" the whole Judaism as a religion thing, but prefers to practice Buddhism, I asked him how it was that he still identified as Jewish. His reply was very telling: "Well, my mom was Jew, and that's how it's passed down, and it's neat to have another culture in my background...plus, I can go to Israel whenever I want."
I don't recall any of the other "bi-racial" people I have known describing their background as "neat" because it gets them automatic citizenship in a country they otherwise don't care about. Usually it's the other way around, where you're too X to be Y, but not X enough to be X.