You're actually looking at this all backwards. Consider this a very high-level answer, as I am an amateur, having researched academic Jewish anthropology only in passing as part of theological research; if you want technical, precise answers to your questions, maybe others can help.

From Jews' perspectives, particularly from the Ashkenazi Jews' perspective, your real name is actually your Hebrew name. Other Jews generally don't have separate English names (although recently this has changed, especially as non-Ashkenazis have moved to the West). Let me elaborate.

Until the mid 18th century (and by most, to this day), Jews were viewed by themselves and others as a nation, not a religion - as Jews, Hebrews, Israelites, etc; separate and apart from the true "European" population. Whether or not a Jew was genuinely religious, Jewish society was Halacha-based and Jewish law meant that Jewish culture and religious observance were intertwined, outside of mainstream European culture. Their heroes were Moses and Queen Esther, not King Richard and Joan of Arc, if that helps you understand. That different culture includes lifestyle, style, and relevant to your question, names.

In the mid-18th century in Central and Eastern Europe, an internal Jewish conflict arose called the Haskala ("education", but more figuratively "the Enlightenment"). Proponents of the Haskala made the central assumption that Jews were bringing their own persecution on themselves by being so different, and an effort was made to be more like the "Europeans". Led by Moses Mendelssohn, some Jews began to assimilate into European culture through art, music and public professions. This is also, in distant origin, where the Reform branch of Judaism comes from - though that is two centuries later and though it is not particularly related to your question, it might help contextualized the Haskalah in something you're probably aware of.

Of course, if Jews were going to be more like their "white" or "European" co-citizens, they needed to look like White Europeans and have White European names for performances, for professional use. If you had a Jewish name, you would be obviously shunned and your professional prospects significantly worsened. So, Jews began to "Europeanize" their names - first, adopting European equivalents for professional use (such as Moshe presenting himself as Moses).

As more and more Jews began to assimilate into European culture, names became more and more European. But those that wanted to hold on to some vestige of their Jewish culture and religious observance, wanted to retain proper Hebrew names. It has since became commonplace for non-religious Jews to have separate names.

Of course, for all of the efforts at assimilation, the next couple centuries in Europe were not great to Jews.

That said, to most Jews, even non-Orthodox, your Hebrew name is your real name. Your local name, which went on your birth certificate for the ostensibly non-Jewish local government, was only so you could navigate the non-Jewish world with fewer roadblocks. At some point, you or your parents may have placed less emphasis on your Jewish identity, and your Hebrew name became a "side" name.

With respect to your second point, Zalman is in fact a longstanding name; the most famous holder of it is Schneur Zalman of Liadi, founder of Chabad Lubavitch. Most Hebrew names are Biblical in origin, with exceptions like Chaya or Chaim, and many Hebrew names are simply the Hebrew forms of non-Hebrew biblical names. The other big difference is that they are usually patronymic. For example, if your dad's Hebrew name was Yaakov your name would be Shlomo Ben Yaakov.

Two very small notes. First, in Ashkenazi custom, it is very very common to name children after dead relatives or important Jewish people (but never living relatives). In Sephardi custom, it is very common to name children after living relatives (and relatively rarely after living relatives). I'm named after my grandfather, for example, who died before I was born.

Second, in further regards to /u/Gamblor29's answer, Sephardis never had the same experience of the Haskalah, so there was less splitting into a public (assimilated with a English/French/German/Polish name) and private (Jewish with a "Hebrew" name that could well be Yiddish, etc). Before emancipation, there were no "neutral communities" in Europe. There was no "just normal" everything was Christian or Jewish. It wasn't even hypothetically possible to assimilate without converting. If I had to guess, the first "non-Hebrew names" were names that Jews gave themselves when dealing with others. Eventually, I assume parents assigned these names at birth. From my experience of minority communities in Turkey (not just Jewish but also Armenian), some people will use one name "in" the community, and one name out in public. For example, I have a friend Sarkiz (which is a very Armenian name) who often goes by Serkan (which is a very Turkish name) in public. You see the same thing with other minority and immigrant communities. It's not rare among East Asian immigrants to America, for example, and Hong Kong and Singapore--places where Chinese and English are both widely used--it's common for people to have both Chinese and English names.

As for contemporary Sephardi practices, Sephardi Jews were deeply influenced by the Alliance schools that opened up all over the Muslim world. Jews I know from Turkey and Morrocco tend to have "European-style" names (especially the women), though some also have Hebrew or local Turkish/Arabic names. To pick prominent Jewish Turkish academics, brother and sister Henri and Karen Barkey have European names, Soli (from Solomon) Özel has a traditional Hebrew names, and Yael Navaro-Yashin has a modern Hebrew name (i.e. it became popular with the revival of Hebrew). In my experience, female Jews in Turkey who have a European or Turkish name don't necessarily think of also having a Hebrew name, though they all know what grandmother they were named after and what her traditional (Hebrew or Ladino) name was. I get the impression that the men have a traditional name that is used to call them to the Torah. The current chief rabbi is named Ishak (the Turkish version of "Isaac"; I bet he's called to the Torah as Yiztak rather than Ishak however) and the previous chief rabbi was name David (rather than the Turkish Davud). The greatest Sephardi sage of the modern era, Ovadia Yosef, was known in Baghdad as Abdullah Youssef, though I imagine his whole life he was called to the Torah as Ovadia ben (ben=son of) Yaakov, rather than Abdullah ben Yaakov (his father's name was Yaakov ben Ovadia, so he was clearly named after his paternal grandfather). Anyway, having a local name was common in Mizrahi lands (look at the Jewish mbrothers, Saleh and Daoud Al-Kuwaity, who were important musicians). Saleh was also the name of the main character in one of the most important early Mizrahi movies, though I don't know how they'd be called up to the Torah--quite possibly as Saleh.

Obviously, most Sephardim today live in Israel, so their Hebew name is generally just their name. Like Ashkanezim, many of them have Hebrew names that are not traditional, and only date to the revival of Hebrew in the 19th century. Still, it's traditional in many places in the post-enlightenment diaspora to have a "Hebrew name" if you have a local language name, but it's not required by halacha. Indeed, many of the "Hebrew names" that Ashkenazi people have are Yiddish. This is very common for women, but less common for men. Still, it happens. Ber (bear in Yiddish), Feivel (ultimately from the Greek Phoebus), Hershel (little deer), Lieber (lover), Selif (happy), Sender (from the Greek for Alexander) were all common names "from the Old Country" that could be used as "Hebrew names" even though they were not Hebrew. With women, it's even more common. /u/gingerkid1234 said a little something about different types of contemporary Jewish naming traditions that's mostly unrelated, but you might still find interesting. He also talks a little about Yiddish double names here.

Also, if you have other Jewish questions, /r/Judaism is a good place to ask--there's even a weekly no such thing as a stupid question thread.